Documentation

Computing L-values

Once an L-series L(s) has been constructed using either a standard L-function (Section Built-in L-series), a user defined L-function (Section Constructing a General L-Series) or constructed from other L-functions (Section Arithmetic with L-series), Magma can compute values L(s₀) for complex s₀, values for the derivatives L^(k)(s₀) and Taylor expansions.

Evaluate(L, s0) : LSer, FldComElt -> FldComElt

    Derivative: RngIntElt               Default: 0

    Leading: BoolElt                    Default: false

Given the L-series L and a complex number s₀, the intrinsic computes either L(s₀), or if D>0, the value of the derivative L^(D)(s₀). If D>0 and it is known that all the lower derivatives vanish, L(s₀)=L'(s₀)=...=L^{(D - 1)}(s₀)=0 , the computation time can be substantially reduced by setting Leading:=true. This is useful if it is desired to determine experimentally the order of vanishing of L(s) at s₀ by successively computing the first few derivatives.

CentralValue(L) : LSer -> FldComElt

Given an L-function of motivic weight 2k - 1, the value of L is computed at s=k.

LStar(L, s0) : LSer, FldComElt -> FldComElt

    Derivative: RngIntElt               Default: 0

Given the L-series L and a complex number s₀, the intrinsic computes either the value L^*(s₀) or, if D>0, the value of the derivative L^{* (D)}(s₀). Here L^*(s)=γ(s)L(s) is the modified L-function that satisfies the functional equation (cf. Section Terminology)
L^*(s) = (sign) .bar L^*((weight) - s)
(cf. Section Terminology).

LTaylor(L,s0,n) : LSer, FldComElt, RngIntElt -> FldComElt

    ZeroBelow: RngIntElt                Default: 0

Compute the first n + 1 terms of the Taylor expansion of the L-function about the point s=s₀, where s₀ is a complex number:
L(s₀) + L'(s₀)x + L"(s₀)x²/2! + ... + L⁽ⁿ⁾(s₀)xⁿ/n! + O(x^{n + 1}) .
If the first few terms L(s₀), ..., L^(k)(s₀) of this expansion are known to be zero, the computation time can be reduced by setting ZeroBelow:=k+1.

Example `Lseries_lseries-evaluate (H136E18)`

We define an elliptic curve E of conductor 5077 and compute derivatives at s=1 until a non-zero value is reached:

> E := EllipticCurve([0, 0, 1, -7, 6]);
> L := LSeries(E : Precision:=15);
> Evaluate(L, 1);
0.000000000000000
> Evaluate(L, 1 : Derivative:=1, Leading:=true);
1.87710082755801E-24
> Evaluate(L, 1 : Derivative:=2, Leading:=true);
-6.94957228421048E-24
> Evaluate(L, 1 : Derivative:=3, Leading:=true);
10.3910994007158

This suggests that L(E, s) has a zero of order 3 at s=1. In fact, E is the elliptic curve (over Q) of smallest conductor with Mordell-Weil rank 3:

> Rank(E);
3

Consequently, a zero of order 3 is predicted by the Birch--Swinnerton-Dyer conjecture. We can also compute a few terms of the Taylor expansion about s=1, with or without specifying that the first three terms vanish.

> time LTaylor(L, 1, 5 : ZeroBelow:=3);
1.73184990011930*$.1^3 - 3.20590558844390*$.1^4 + 2.80009237167013*$.1^5 +
   O($.1^6)
Time: 0.800
> time LTaylor(L, 1, 5);
1.87710082755801E-24*$.1 - 3.47478614210524E-24*$.1^2 + 1.73184990011930*$.1^3
   - 3.20590558844390*$.1^4 + 2.80009237167013*$.1^5 + O($.1^6)
Time: 1.530

And this is the leading derivative, with the same value as Evaluate(L,1:D:=3).

> c := Coefficient($1,3)*Factorial(3);c;
10.3910994007158

Finally, we compute the 3rd derivative of the modified L-function L^*(s)=γ(s)L(s) at s=1. For an elliptic curve over the rationals, γ(s)=(N/π²)^s/2Γ(s/2)Γ((s + 1)/2), where N is the conductor. So, by the chain rule, (L^*)"'(1)=γ(1)L"'(1)=Sqrt(N/π)L"'(1).

> LStar(L, 1 : Derivative:=3);
417.724689268266
> c*Sqrt(Conductor(E)/Pi(RealField(15)));
417.724689268267