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We prove that more than nine percent of the central values$L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges overprimes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$.Previously, it was not known whether a positive proportion of these centralvalues are non-zero. As a by-product, we obtain the order of magnitude of thesecond moment of $L(\frac{1}{2},\chi_p)$, and conditionally we obtain the orderof magnitude of the third moment. Assuming the Generalized Riemann Hypothesis,we show that our lower bound for the second moment is asymptotically sharp.

Dirichlet $L$-functions of quadratic characters of prime conductor at the central point