Analytic continuation of primitive Dirichlet L-functions

Mellin-transform ${\sum }_{n\ge 1}\chi (n)n^a{e}^{-\pi n^{2}t\mathrm{/}q}$; for $\mathrm{Re}s>1$ this equals a nonzero constant times $\mathrm{\Lambda }(s,\chi )$.

Apply [F1] in [F2] to transform the theta kernel under $t\mapsto 1\mathrm{/}t$. Splitting the Mellin integral at $1$ gives two rapidly decaying integrals, hence an entire continuation when $\chi$ is nonprincipal.

The only primitive principal case is $q=1$. Splitting the Mellin integral for the Jacobi theta kernel and separating its constant term continues ${\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)\zeta (s)$ meromorphically, with the two boundary terms giving simple poles at $s=0$ and $s=1$. Equivalently, the standard entire completion is $\frac{1}{2}s(s-1){\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)\zeta (s)\mathrm{.}$ Fact [F3] separately says that $L=\zeta$ itself has only its simple pole at $s=1$.