Background: I'm facing the computation of the zeta regularization of the infinite product given by
$$\prod_{m=-\infty}^\infty (km+u)$$
for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. H. Heydari and R. Y. Song (Example 10) I know that the zeta regularization of $\prod_{m=-\infty}^\infty (m+u)$ is given by
$$\prod_{m=-\infty}^\infty (m+u) = \begin{cases}1-e^{2\pi i u} & \Im(u) > 0 \\ 1-e^{-2\pi i u} & \Im(u) < 0\end{cases}$$
so a possible reasoning is as follows:
$$\prod_{m=-\infty}^\infty (km+u) = \prod_{m=-\infty}^\infty k(m+uk^{-1})$$
Now, using formula (1) from J. R. Quine, S. H. Heydari and R. Y. Song we have that
$$\prod_{m=-\infty}^\infty k(m+uk^{-1}) = k^{Z(0)}\prod_{m=-\infty}^\infty (m+uk^{-1})$$
and we are reduced to computing $Z(0)$ where $Z(s)$ is the analytic prolongation of $Z(s)=\sum_{m \in \mathbb{Z}}(m+uk^{-1})^{-s}$. We can write $Z(s)$ as
$$Z(s) = \sum_{m > 0}(m+uk^{-1})^{-s}+(uk^{-1})^{-s}+\sum_{m <0}(m+uk^{-1})^{-s}$$
By adding and subtracting the $m=0$ term to both of the series and by changing variable from $m$ to $-m$ in the sum indexed by negative integers we get
$$\begin{align} Z(s) &= \sum_{m=0}^\infty(m+uk^{-1})^{-s} - (uk^{-1})^{-s} + \sum_{m=0}^\infty(uk^{-1}-m)^{-s} \\ &= \sum_{m=0}^\infty(m+uk^{-1})^{-s} - (uk^{-1})^{-s} + e^{\pi i s}\sum_{m=0}^\infty(m-uk^{-1})^{-s} \\ &= \zeta(s,uk^{-1}) -(uk^{-1})^{-s} + e^{\pi is}\zeta(s,-uk^{-1}) \end{align}$$
where $\zeta(s,a)$ is the Hurwitz Zeta function. So that by using formula 25.11.13 again in https://dlmf.nist.gov/25.11#E13 we found that
$$Z(0) = \dfrac{1}{2}-uk^{-1}-1+\dfrac{1}{2}+uk^{-1}=0$$
and we end up with
$$\prod_{m=-\infty}^\infty k(m+uk^{-1}) = k^{Z(0)}\prod_{m=-\infty}^\infty (m+uk^{-1}) = \prod_{m=-\infty}^\infty (m+uk^{-1}) = \begin{cases}1-e^{2\pi i uk^{-1}} & \Im(uk^{-1}) > 0 \\ 1-e^{-2\pi i uk^{-1}} & \Im(uk^{-1}) < 0\end{cases}$$
Question: I'm not confident enough with these manipulations involving analytic prolongations so I may be wrong in the above derivation of $Z(0)=0$, Is the above result correct? Does it remain valid if one replaces the condition $k$ real and positive with $k$ a nonzero complex number?