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 extension of $G$, let $K_p$ be the completion at $p$. If $T$ is the maximal central torus in $G \otimes_K K_p$, 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) w.r.t. to the local component of the Tamagawa measure. This would make the infinite product converge. In the number field case, this volume is $$ |k|^{-d} \cdot |\overline{T}(k)| =\det(1_d-h(F)/|k|) = L_p(1,\chi_T)^{-1}. $$ where $k$ is the residue field, $\overline{T}$ is the special fiber of the NR-finite type model of $T$, $d= \dim T$, $h(F)$ is the image of the Frobenius automorphism in the character module representation of the Galois group and $\chi_T$ is the character of the represnetation.
For some reason which I cannot understand,
J. Oesterle in "Nombres de Tamagawa et groupes unipotents en caractéristique $p$"
section 2.5 has found that the volume w.r.t. the local component in the Tamagawa measure -
in our function field case - is: $q^{-n-d} \cdot |\overline{G}(k)|$.
It is not clear what is $n$.
I saw that Kai Behrend and Ajneet Dhillon in their paper
"The geometry of Tamagawa numbers of Chevalley groups"
are quoting in Proposition 6.1 Oesterle saying that $n$ is the degree of the prime $p$.
According to that, the $L$-function should differ from the aforementioned one in the Number field case.
Could someone explain me how and why ?
Thank you, rony.
