Functional equation and/or growth estimates for a shifted L function Consider the $L$-series defined by
$$L_{\alpha,\chi}(s) = \sum_{n\geq 1} \frac{e^{2\pi i \alpha \Omega(n)} \chi(n)}{n^s} = \prod_p \left(1 - \frac{e^{2\pi i \alpha} \chi(p)}{p^s}\right)^{-1}.$$
It should have an analytic continuation the left of $\Re s = 1$.
Does it have any poles other than (possibly) $s=1$? Does it satisfy a functional equation? Or rather (since this is why I want a functional equation): what kind of growth estimates do we have on $L_{\alpha,\chi}(\sigma + i t)$ for fixed $\sigma < 1$ and variable $t$, $|t|\to \infty$?
(I'm sure this function is known and has most likely been studied - I just do not remember where I have ever seen it.)
 A: I dispute that this should be termed an "L-function", but anyway...
Taking logs for $\sigma>1$ you get
$$\log L_{\alpha,\chi}(s)=-\sum_p\sum_k{e(k\alpha)\chi(p^k)\over kp^{ks}}$$
So this is
$$\log L_{\alpha,\chi}(s)=-e(\alpha)\sum_p\sum_k{\chi(p^k)\over kp^{ks}}
-\sum_p\sum_{k=1}^\infty{\chi(p^k)\over kp^{ks}}[e(k\alpha)-e(\alpha)]$$
and as the $k=1$ term in the latter double sum vanishes, this double sum,
call it $G(s)$, is analytic for $\sigma>1/2$, and is bounded for $\sigma\ge \sigma_0$ there.
Exponentiating gives $$L_{\alpha,\chi}(s)=\exp(e(\alpha)\log L_{0,\chi}(s)+G(s))=L(s,\chi)^{e(\alpha)}\exp(G(s))$$
Thus: this gives an analytic continutation of ${L'_{\alpha,\chi}(s)\over L_{\alpha,\chi}(s)}$ to $\sigma>1/2$ away from zeros of $L(s,\chi)$, and similarly for $L_{\alpha,\chi}(s)$ though the zeros will now be branch points.
As for growth, you'd probably want to consider things in terms of the logarithm. For fixed $\sigma<1$, to get a fair-play asymptotic you'd likely have to assume that there are no zeros to the right of $\sigma$ of $L(s,\chi)$, or maybe additionally that $\sigma$ is not a limit point of real parts of zeros.
A: This preprint by Kaczorowski and Perelli may contain the pieces of information you're looking for: https://arxiv.org/abs/1911.10497
