Let G be an algebraic reductive group defined over an algebraic function field K in one variable with a finite field of constants $\mathbf{F}_q$. For any prime p of K, unramified in the Galois splitting group of G, let $K_p$ be the completion at p. If T is the maximal central torus in $G \times_K K_p$, then the local Artin L-function, should be the inverse of the volume of the maximal compact subgroup of $T(K_p)$ (trivial in the semisimple case). This would make the infinite product converge. In the number field case, this volume is $|k|^{-d} \cdot |\overline{T}(k)|$ where k is the residue field, $\overline{T}$ is the special fiber of the NR-finite type model of T and $d= \dim T$.
For some reason which I cannot understand, J. Oesterle has found that the volume w.r.t. the local component in the Tamagawa measure -- in our function field case -- is: $q^{-n-d} |\overline{G}(k)|$. It is not clear what is n. I saw a paper quoting Oesterle saying that n is the degree of the prime p. According to that, the L-function should differ from the one in the Number field case. Could someone explain me how and why ?
Thank you, rony.
