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I have been looking for a proof of the statement: "Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, $$L(\tfrac{1}{2},\chi_d)\geq 0."$$ Can anyone point me in the right direction or give a reference? Thanks a lot!

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up vote 6 down vote accepted

For real $s$, $L(s,\chi_d)$ is real, and it is certainly positive for large $s$. It is nonzero for $s>1$ by the Euler product and $L(1,\chi_d)\ne0$. If $L(1/2,\chi_s)<0$ what would happen for some $1/2<s<1$?

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Very nice answer, thank you. – Troy K. Dec 10 '10 at 19:23

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