Exceptional characters and nonvanishing of Dirichlet L-functions

Abstract

Let \(\psi \) be a real primitive character modulo D. If the L-function \(L(s,\psi )\) has a real zero close to \(s=1\), known as a Landau–Siegel zero, then we say the character \(\psi \) is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values \(L(1/2,\chi )\) of the Dirichlet L-functions \(L(s,\chi )\) are nonzero, where \(\chi \) ranges over primitive characters modulo q and q is a large prime of size \(D^{O(1)}\). Under the same hypothesis we also show that, for almost all \(\chi \), the function \(L(s,\chi )\) has at most a simple zero at \(s = 1/2\).