Product over all Dirichlet characters (${\mathcal{P}}_q(s)={\prod }_{\chi \mathrm{mod}q}L(s,\chi )$)

For real $s>1$, Orthogonality of Dirichlet characters gives $\log {\mathcal{P}}_q(s)=\phi (q){\sum }_{p\nmid q}{\sum }_{j\ge 1,p^j\equiv 1(\mathrm{mod}q)}1/(j{p}^{js})\ge 0$. Thus ${\mathcal{P}}_q(s)\ge 1$. 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.