A Counter-Example in the Theory of Local Zeta Functions

Roland Martin

Abstract

The generalized Igusa local zeta function Z_{Q_{p}}(s) associated to (SL_{7}, ρ), where ρ 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↦p^{−1}} = p^{−7s}Z_{Q_{p}}(s).