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 𝐷𝑂(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.