A note on exceptional characters and non-vanishing of Dirichlet L-functions

We study non-vanishing of Dirichlet L-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if ψ is a real primitive character modulo D ∈ N with L(1,ψ) ≪ (log D)−^25−ε, then, for any prime q ∈ [D³⁰⁰, D^O(1)], one has L(1/2, χ) ≠ 0 for almost all Dirichlet characters χ (mod q).