A $GL_1$ Voronoi formula

Question

I want a functional equation for the function defined by the Dirichlet series, $$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$ which sends $s$ to $1-s$ and preferably sends $a$ to $\bar{a}$ where $a\bar{a} \equiv 1 \pmod q$.

What I want should possibly be classified as a pre-Voronoi formula, where afterwards one takes a Mellin transform.

I could slightly generalize and look for a transformation formula for $$ \sum_{n=1}^\infty \frac{\chi(n) e^{2\pi i n a/q}}{n^s}$$ where $\chi$ is a Dirichlet charater. But I would be perfectly content with a functional equation for the first one as well.

One can write the above Dirichlet series in terms of Hurwitz zeta functions and apply the the functional equation of the Hurwitz zeta functions. When that is done, however, one does not have $D(\cdot,\cdot)$ on the right hand side but Hurwitz zeta functions, and neither does it change $a$ to $\bar{a}$.

Is such a formula known already?

Answer 1

Here is one version of the desired functional equation: let $a,q$ be positive integers with $(a,q) = 1$, and define the Dirichlet series $$\Phi(q,a;s) = \sum_{n = 1}^{\infty} \frac{e\left(\frac{an}{q}\right)}{n^s}.$$ This converges absolutely for $\Re(s) > 1$. It extends meromorphically to $\mathbb{C}$; it is entire unless $q = 1$, in which case there is a simple pole at $s = 1$ with residue $1$. It satisfies the functional equation $$\Phi(q,a;s) = \sum_{\pm} G^{\mp}(1 - s) \Xi(q,\pm a;-s),$$ where $$G^{\pm}(s) = (2\pi)^{-s} \Gamma(s) \exp\left(\pm \frac{i\pi s}{2}\right)$$ and $$\Xi(q,\pm a;-s) = q^{-s} \sum_{b \in \mathbb{Z}/q\mathbb{Z}} e\left(\pm \frac{ab}{q}\right) \sum_{m = 1}^{\infty} \frac{e\left(\frac{bm}{q}\right)}{m^{1 - s}} = q^{1 - s}\sum_{\substack{m = 1 \\ m \equiv \mp a \pmod{q}}}^{\infty} \frac{1}{m^{1 - s}}.$$ Note that unlike the $\mathrm{GL}_2$ Voronoi summation formula, the "dual" series in this functional equation involves $\pm a$ rather than $\pm \overline{a}$ (where $\overline{a}$ denotes the multiplicative inverse of $a$ modulo $q$).

One can prove this simply by using the functional equation for the Hurwitz zeta function, or writing $e\left(\frac{an}{q}\right)$ in terms of a sum over Dirichlet characters and using the functional equation for (possibly imprimitive) Dirichlet $L$-functions.