Motives (2/3)
The local Euler factor at a good prime p is then given by the (inverse of the) usual formula
Pp(T) = exp

−
X
f≥1
Npf (H, t)
f
Tf

 ,
always a polynomial of degree d. N.B. the Euler factor Lp used in the global L-function is
1/Pp(p−s). The formulas is modified for tame primes or for t = 1 (Roberts, Rodriguez Villegas,
Watkins,. . . ) and usually deg Pp < d in this case.
Various recipes are conjectured for wild primes (often Lp ≡ 1) but we did not implement them.
On the other hand Lp can be guessed via the global functional equation: once a global L-function
is computed, we can obtain Euler factor at any prime, using lfuneuler.
Atelier 2022 (13/01/2022) – p. 8/12