A counterexample in the theory of local zeta functions

Abstract

The generalized Igusa local zeta function $Z_{\Q _p}(s)$ associated to $(\SL_7,\rho)$, where $\rho$ is the Cartan product of the first, third and fifth fundamental representations of $\SL_7$, is explicitly computed and shown not to satisfy the expected functional equation $$Z_{\Q _p}(s)|_{p \mapsto p^{-1}}=p^{-7s}Z_{\Q _p}(s).$$