Small zeros of Dirichlet L-functions of quadratic characters of prime modulus

Abstract

In this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet L-functions associated to the quadratic characters χp(⋅)=(⋅|p) with p a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of L-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of (−1,1). We also write down the ratios conjecture for this family of L-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz–Sarnak prediction and includes lower order terms. Following the methods of Özlük and Snyder, we prove that GRH implies L(12,χp)≠0 for at least 75% of the primes.