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 $\psi$ is a real primitive character modulo $D\in \mathbb{N}$ with $L(1,\psi )\ll (\log D{)}^{-25-\epsilon }$, then, for any prime $q\in [D^{300},D^{O(1)}]$, one has $L(1\mathrm{/}2,\chi )\mathrm{\ne }0$ for almost all Dirichlet characters $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$.