Exceptional characters and nonvanishing of Dirichlet L-functions

Let ψ be a real primitive character modulo D. If the L-function L(s, ψ) has a real zero close to s= 1 , known as a Landau–Siegel zero, then we say the character ψ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values L(1 / 2 , χ) of the Dirichlet L-functions L(s, χ) are nonzero, where χ 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 χ, the function L(s, χ) has at most a simple zero at s= 1 / 2.