Dirichlet characters and low-lying zeros of L-functions

Let r be a positive integer >= 2. We consider a family of primitive Dirichlet characters of order r with conductor co-prime to r. For this family, we compute the one-level density with explicit lower order terms in two ways, using Weil's explicit formula and the Ratios conjecture. Also, the n. level density for the family twisted by a fixed cuspidal automorphic representation pi of GL(M) (A(Q)) is obtained. It turns out that, when r >= 3, the symmetry type for our family is always unitary. (C) 2019 Elsevier Inc. All rights reserved