Euler Product and Determinant Local $L$-Factor

An Euler product is a product over primes (or places) of local factors, typically convergent for $\mathrm{Re}(s)\gg 0$.

In the letter, given a representation $\pi$ of the $L$-group and Satake parameters $\alpha_p$, the unramified local factor is $L_p(s)=\det\!\bigl(1-\pi(\alpha_p)\,p^{-s}\bigr)^{-1}.$

The global $L$-function is $L(s)=\prod_p L_p(s)$, with finitely many “bad primes” omitted or modified.

Key point: changing auxiliary choices typically changes only finitely many local factors.