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 .