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<[email protected]> |
Dear Karim,
Many thanks for your comprehensive response. All clear now!
Cheers,
Rudolph
On 30/12/2023, 15:35, "Karim Belabas" <[email protected]> wrote:
Dear Rudolph,
as documented in ??13 (or ?? "L-function"), only "primitive" L-functions
are supported by PARI's implementation.
In particular for Dirichlet L-function (and more generally Hecke
L-functions), a character given to any modulus encodes the attached
*primitive* character. Thus all principal characters will yield the
L-function attached to the trivial character mod 1, i.e., the Riemann
zeta function.
The Dirichlet L-function attached to the actual non-primitive character
mod N differs from the primitive one (of conductor F) by a simple finite
Euler product
\prod_{p | N, p \nid F} (1 - \chi(p) p^{-s})
Just multiply the value returned by lfun() by this factor
(you may precompute the \chi(p)...).
If you're only interested in principal characters mod N, this is as
simple as
P = factor(N)[1,]; \\ can be precomputed
ZetaN(P, s) = zeta(s) * prod(j = 1, #P, 1 - P[j]^(-s));
Cheers,
K.B.
* [email protected] [2023-12-30 15:04]:
> Dear developers,
>
>
> I’d like to generate Dirichlet L-functions and used for instance:
>
>
> default(realprecision,30)
>
> p = 2; q = 3;
>
> L =lfuncreate(Mod(p,q));
>
> print(lfun(L,2));
>
>
> This method works well for all non-principal characters, however seems
> to fail for the principal ones with q > 1:
>
>
> default(realprecision,30)
>
> p = 1; q = 3;
>
> L =lfuncreate(Mod(p,q));
>
> print(lfun(L,2));
>
>
> 1.64493406684822643647241516665 = (pi^2)/6
>
> which should be:
>
> 1.46216361497620127686436903702 = (4*pi^2)/27
>
>
> It always seems to default to the zeta-function even when the modulus
> is greater than 1.
>
>
> Maybe I do something wrong here and is the Mod(p,q) not allowed for
> principal characters. Keen to learn how to obtain the right outcome.
>
>
> Thanks,
>
> Rudolph
--
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/