Recently, I have been pushed toward studying analytic continuation dirichlet series with twists that are additive. These are functions $D(s) = \sum e_k(hn)\frac{a_n}{n^s}$ where $a_n$ is some sequence (usually positive) and $e_k(n)=e^{2\pi i \frac{n}{k}}$ with $h\in \mathbb{N}.$
I would like to know if there is any literature on this topic available.