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The Euler product for $L(s,\chi)$ will tell you that $L(\sigma,\chi)>0$ for real $\sigma>1$. Moreover, $L(s,\chi)$ is continuous....

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Although there might be some trouble in implementing this in practice, I had found a trick that same afternoon that seemed to work. I defined this finite difference operator: $$\square_{a,b} F(s) = \frac{F(s-1)-bF(s)}{a-b}$$ It "kills" the $b$th term in the Dirichlet series while leaving the $a$th term untouched. In order to extract the term I want, I do ...

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By Kronecker's formula, the value at $s=1$ of $L(s,\chi)$ functions associated with quadratic characters depends on a class number, so there actually is an algebraic counterpart dealing with reduced binary quadratic forms. The basic problem is to find the sign of a Gauss sum, where $G(\chi)^2 = p$ is almost trivial. However, I am not so sure that "the ...

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