The global L-function (1/2)
If one suitably defines Pp(T) for all primes p including the wild ones, then the L-function defined
by
L(H, s) =
Y
p
Pp(p−s
)−1
is motivic (Katz), with analytic continuation and functional equation, as used in the L-function
package of Pari/GP. If the motivic weight w is even, there is a possible (multiple) pole at
w/2 + 1.
The command L = lfunhgm(H, t) creates such an L-function. In particular it must guess
the local Euler factors at wild primes, which can be very expensive when the conductor
lfunparams(L)[1] or the degree d is large. This L-function can then be used with all the
functions of the lfun package. For instance we can now obtain the global conductor and check
the Euler factors at all bad primes.
In our example, lfunhgm(H, 1/2) is very fast (only 5 is wild and the conductor is 5000).
More complicated, L = lfunhgm(H, 1/64) finishes in about 20 seconds (the conductor is
525000).
Atelier 2022 (13/01/2022) – p. 10/12