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Thursday, October 4, 2018

**Abstract:** In this joint work with Kyle Pratt, we prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1$ (mod $8$) ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain estimates for the second and third moments of $L(\frac{1}{2},\chi_p)$.