n-Level Densities of Artin L-Functions

Abstract

In this paper, we consider a family of twisted Artin L-functions, L(s, pi X rho), where 7 is a fixed self-dual cuspidal representation of GL, and p is given by L(s, rho, K) = sigma(s)/sigma(s) attached to an Sd+1-field K. By the strong Artin conjecture, we consider p as a cuspidal representation of GL(d). We obtain n-level densities for our families under certain counting conjectures. Our result is unconditional for S-3-fields regardless of 7, which is of symplectic type or of orthogonal type. For 7t of orthogonal type (i.e., the symmetric square L-function has a pole at s = 1), the n-level density computation is unconditional for S-4-fields (and S-5-fields under the strong Artin conjecture for rho)