The zeta regularization of $prod_{m=-infty}^infty (km+u)$

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

$$prod_{m=-infty}^infty (m+u) = begin{cases}1-e^{2pi i u} & Im(u) > 0 \ 1-e^{-2pi i u} & Im(u) < 0end{cases}$$

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

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} + (-1)^{-s}sum_{m=0}^infty(m-uk^{-1})^{-s} \
&= zeta(s,uk^{-1}) -(uk^{-1})^{-s} + (-1)^{-s}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 we found that

$$Z(0) = dfrac{1}{2}+uk^{-1}-1+dfrac{1}{2}-uk^{-1}=0$$

and we end up with

$$prod_{m=-infty}^infty k(m+uk^{-1}) = k^{Z(0)}prod_{m=-infty}^infty (m+uk^{-1}) = prod_{m=-infty}^infty (m+uk^{-1}) = begin{cases}1-e^{2pi i uk^{-1}} & Im(uk^{-1}) > 0 \ 1-e^{-2pi i uk^{-1}} & Im(uk^{-1}) < 0end{cases}$$

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?