Central non-vanishing of Dirichlet $L$-functions

We prove that, for every sufficiently large modulus $q\equiv ̸24$, at least $3\mathrm{/}8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1\mathrm{/}2,\chi )\mathrm{\ne }0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.