Re: Dirichlet L series principal characters

Dear Karim,

Many thanks for your comprehensive response. All clear now!

Cheers,

Rudolph

On 30/12/2023, 15:35, "Karim Belabas" <Karim.Belabas@math.u-bordeaux.fr> 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.

* ra.dwars@quicknet.nl [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/