The smallest prime in a conjugacy class and the first sign change for automorphic 𝐿-functions

Abstract

Let K K be an S n S_n -field. For a nonidentity conjugacy class C C , define N K , C N_{K,C} to be the smallest prime p p such that Frob p ∈ C _p\in C . By using the observation that N K , C N_{K,C} is interpreted as the first prime sign change of the Dirichlet coefficients of automorphic L L -functions, we improve the known bound on N K , C N_{K,C} for n = 3 , 4 , 5 n=3,4,5 . (For n = 5 n=5 , we need to assume the strong Artin conjecture.)