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Dokchitser’s L-functions Calculator

AUTHORS:

  • Tim Dokchitser (2002): original PARI code and algorithm (and the documentation below is based on Dokchitser’s docs).
  • William Stein (2006-03-08): Sage interface

TODO:

  • add more examples from SAGE_EXTCODE/pari/dokchitser that illustrate use with Eisenstein series, number fields, etc.
  • plug this code into number fields and modular forms code (elliptic curves are done).
class sage.lfunctions.dokchitser.Dokchitser(conductor, gammaV, weight, eps, poles=[], residues='automatic', prec=53, init=None)

Bases: sage.structure.sage_object.SageObject

Dokchitser’s \(L\)-functions Calculator

Create a Dokchitser \(L\)-series with

Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec)

where

  • conductor - integer, the conductor
  • gammaV - list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples).
  • weight - positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for \(H^1\) of curves/\(\QQ\)
  • eps - complex number; sign in functional equation
  • poles - (default: []) list of points where \(L^*(s)\) has (simple) poles; only poles with \(Re(s)>weight/2\) should be included
  • residues - vector of residues of \(L^*(s)\) in those poles or set residues=’automatic’ (default value)
  • prec - integer (default: 53) number of bits of precision

RIEMANN ZETA FUNCTION:

We compute with the Riemann Zeta function.

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
Traceback (most recent call last):
...
ArithmeticError
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)

RANK 1 ELLIPTIC CURVE:

We compute with the \(L\)-series of a rank \(1\) curve.

sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0.000000000000000
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation()
6.11218974700000e-18                            # 32-bit
6.04442711160669e-18                            # 64-bit

RANK 2 ELLIPTIC CURVE:

We compute the leading coefficient and Taylor expansion of the \(L\)-series of a rank \(2\) curve.

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs ()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
2.90760251490292e-20 + (-1.64772944938078e-20)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)  # 32-bit
-3.11661104824958e-21 + (1.76617394576638e-21)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)  # 64-bit

RAMANUJAN DELTA L-FUNCTION:

The coefficients are given by Ramanujan’s tau function:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)

We redefine the default bound on the coefficients: Deligne’s estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster).

sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11

Now we’re ready to evaluate, etc.

sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
check_functional_equation(T=1.2)

Verifies how well numerically the functional equation is satisfied, and also determines the residues if self.poles != [] and residues=’automatic’.

More specifically: for \(T>1\) (default 1.2), self.check_functional_equation(T) should ideally return 0 (to the current precision).

  • if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong),
  • if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata(“a(k)”,1.3), checkfeq(1.3)
  • T=1 always (!) returns 0, so T has to be away from 1
  • default value \(T=1.2\) seems to give a reasonable balance
  • if you don’t have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata(“a(k)”,1), you need slightly less coefficients.

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-1.35525271600000e-20                        # 32-bit
-2.71050543121376e-20                        # 64-bit

If we choose the sign in functional equation for the \(\zeta\) function incorrectly, the functional equation doesn’t check out.

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-9.73967861488124
derivative(s, k=1)

Return the \(k\)-th derivative of the \(L\)-series at \(s\).

Warning

If \(k\) is greater than the order of vanishing of \(L\) at \(s\) you may get nonsense.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.derivative(1,E.rank())
1.51863300057685
gp()

Return the gp interpreter that is used to implement this Dokchitser L-function.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter
init_coeffs(v, cutoff=1, w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40)

Set the coefficients \(a_n\) of the \(L\)-series. If \(L(s)\) is not equal to its dual, pass the coefficients of the dual as the second optional argument.

INPUT:

  • v - list of complex numbers or string (pari function of k)
  • cutoff - real number = 1 (default: 1)
  • w - list of complex numbers or string (pari function of k)
  • pari_precode - some code to execute in pari before calling initLdata
  • max_imaginary_part - (default: 0): redefine if you want to compute L(s) for s having large imaginary part,
  • max_asymp_coeffs - (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t).

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)

Evaluate the resulting L-function at a point, and compare with the answer that one gets “by definition” (of L-function attached to a modular form):

sage: L(14)
0.998583063162746
sage: a = delta_qexp(1000)
sage: sum(a[n]/float(n)^14 for n in range(1,1000))
0.9985830631627459

Illustrate that one can give a list of complex numbers for v (see trac 10937):

sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:])
sage: L2(14)
0.998583063162746

TESTS:

Verify that setting the \(w\) parameter does not raise an error (see trac 10937). Note that the meaning of \(w\) does not seem to be documented anywhere in Dokchitser’s package yet, so there is no claim that the example below is meaningful!

sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000])
num_coeffs(T=1)

Return number of coefficients \(a_n\) that are needed in order to perform most relevant \(L\)-function computations to the desired precision.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
set_coeff_growth(coefgrow)

You might have to redefine the coefficient growth function if the \(a_n\) of the \(L\)-series are not given by the following PARI function:

coefgrow(n) = if(length(Lpoles),
                  1.5*n^(vecmax(real(Lpoles))-1),
                  sqrt(4*n)^(weight-1));

INPUT:

  • coefgrow - string that evaluates to a PARI function of n that defines a coefgrow function.

EXAMPLE:

sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155
taylor_series(a=0, k=6, var='z')

Return the first \(k\) terms of the Taylor series expansion of the \(L\)-series about \(a\).

This is returned as a series in var, where you should view var as equal to \(s-a\). Thus this function returns the formal power series whose coefficients are \(L^{(n)}(a)/n!\).

INPUT:

  • a - complex number (default: 0); point about which to expand
  • k - integer (default: 6), series is \(O(``var``^k)\)
  • var - string (default: ‘z’), variable of power series

EXAMPLES:

sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser()
sage: L.taylor_series(1)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)

We compute a Taylor series where each coefficient is to high precision.

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200)
sage: L.taylor_series(1,3)
-9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)
sage.lfunctions.dokchitser.reduce_load_dokchitser(D)

x.__init__(...) initializes x; see help(type(x)) for signature

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