Let $K$ be a (real quadratic) number field and $\chi$ be a Hecke character on it. We can think about Hecke $L$-function $$ L(\chi, s):= \sum_{0\neq \mathfrak{a}\subseteq \mathcal{O}_{K}} \frac{\chi(\mathfrak{a})}{(N\mathfrak{a})^{s}} = \sum_{n\geq 1} \frac{a(n)}{n^{s}} $$ which has analytic continuation on $\mathbb{C}$ and admits a functional equation.
Now I'm interested in the following "twisted" Hecke $L$-function: $$ L(\chi, \beta, s):= \sum_{n\geq 1} \frac{\beta^{n}\cdot a(n)}{n^{s}} $$ where $\beta$ is a root of unity. Does this $L$-function have analytic continuation and functional equation? I discovered some identities related to these $L$-values at negative integers, and the identities are completely nonsense if $L(\chi,\beta, s)$ doesn't have analytic continuation.
More precisely, I'm interested in the case when Hecke character $\chi$ of $\mathbb{Q}(\sqrt{2})$ defined as $$ \chi(\mathfrak{a}) = \begin{cases} 1 & N\mathfrak{a}\equiv \pm 1\mod{16} \\ -1 & N\mathfrak{a}\equiv \pm 7 \mod{16} \\ 0 & otherwise \end{cases} $$ for any $\mathfrak{a}\subset \mathbb{Z}[\sqrt{2}]$, and $\beta =\zeta_{8p}$ for any odd prime $p$. I hope there is a formula for the value $L(\chi, \zeta_{8p}^{j}, -n)$ for $n>0$ and $j=0, \dots, p-1$.
Yes, this has an analytic continuation and functional equation. Basically, the crucial point is that the sequence $\{a(n)\}$ is equal to the sequence of Hecke eigenvalues of an automorphic form. Then the twist by a root of unity of the associated $L$-function appears in the Voronoi summation formula (sometimes this twisted $L$-function is called an Estermann $L$-function).
The functional equation is quite complicated; it depends highly on the level of the associated automorphic form, the data appearing in the gamma factors of the untwisted $L$-function, and the exact choice of root of unity. A starting point is Appendix A of the paper of this paper.