Let $F$ be an algebraic number field. Let $\gamma=e(2 \pi i \alpha)$ be. Define the following Lerch zeta function
$$ \zeta_{F}(s, \gamma)=\sum_{\mathfrak{a} \subset \mathcal{O}_{F}} \frac{\gamma^{N(\mathfrak{a})}}{N(\mathfrak{a})^s} $$
where $\mathfrak{a}$ runs over the set on non-zero integral ideals in the ring of algebraic integers $\mathcal{O}_{F}$ and $N(\mathfrak{a})$ is the norm of $\mathfrak{a}$. When $F=\mathbb{Q}$, $\zeta_{F}(s,\gamma)$ is the classical Lerch zeta function $\zeta(s,1,\gamma)$.
When $\gamma=1$, $\zeta_{F}(s,1)$ is nothing but the classical Dedekind zeta function. The Siegel-Klingen theorem says that $\zeta_{F}(-n,1) \in \mathbb{Q}$ for all non-negative integers $n$. When $F=\mathbb{Q}$, we also know that $\zeta_{F}(-n, \gamma) \in \mathbb{Q}(\gamma)$. However, the proofs of these two facts are quite different. While the Siegel-Klingen theorem is proved by the theory of Hilbert modular forms, the second statement can be proved by standard contour integrals techniques.
My question is: is this known that $\zeta_{F}(-n, \gamma) \in \mathbb{Q}(\gamma)$?