|
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
Once an L-series L(s) has been constructed using either a standard
zeta- or L-function (Built-in L-series), a user defined L-function
(Constructing a General L-Series) or constructed from other L-functions (Arithmetic with L-series),
Magma can compute values L(s0) for complex s0, values for
the derivatives L(k)(s0) and Taylor expansions.
Derivative: RngIntElt Default: 0
Leading: BoolElt Default: false
Compute L(s0) or the value of the derivative L(D)(s0) for
D>0 where L is an L-series and s0 is a complex number.
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.
Given a L-function of even weight 2k (in the Magma sense),
compute the value of L at s=k.
Derivative: RngIntElt Default: 0
Compute the value L^ * (s0) or the derivative L * (D)(s0) for D>0,
where s0 is a complex number.
Here L^ * (s)=γ(s)L(s) is the modified L-function that satisfies
the functional equation (cf. Terminology)
L^ * (s) = ( sign) .bar L^ * (( weight) - s)
(cf. Terminology).
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(xn + 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.
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);
-1.69522909186553E-17
> Evaluate(L, 1 : Derivative:=2, Leading:=true);
6.27623031179552E-17
> 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 rational elliptic curve 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 + 2.93970849657696*$.1^5
Time: 11.070
> time LTaylor(L, 1, 5);
-1.69522909186553E-17*$.1 + 3.13811515589776E-17*$.1^2 +
1.73184990011930*$.1^3 - 3.20590558844390*$.1^4 + 2.93970849657696*$.1^5
Time: 20.320
And this is the leading derivative, the same 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
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
