= Product over all Dirichlet characters
{title2=$\mathcal P_q(s)=\prod_{\chi\bmod q}L(s,\chi)$}
For real $s>1$, <Orthogonality of Dirichlet characters> gives $\log\mathcal P_q(s)=\varphi(q)\sum_{p\nmid q}\sum_{j\ge1,\ p^j\equiv1\pmod q}1/(jp^{js})\ge0$. Thus $\mathcal P_q(s)\ge1$. Only the <principal Dirichlet character> contributes a <pole> at one, and that <pole> is simple. A zero of a nonreal <Dirichlet L-function> at one brings a zero of its conjugate too; their two zeros overpower that <pole> and would make the product tend to zero. This proves nonvanishing for nonreal <Dirichlet characters> and is a useful complement to arguments for <real Dirichlet characters>.
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