Non-Vanishing of Dirichlet $L$-functions at the central point with restricted root number

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime conductor $q$, a positive proportion of the central values $L(1/2,\chi)$ do not vanish as $q\to\infty$.