My question considers the regularity of the Epstein zeta function in one of it modules. Let $h\in \mathbb R^d$ and $A\in \mathbb R^{d\times d}$ positive definite (we can also assume $A=I$ for simplicity). We consider the function $$ \mathbb R^d\setminus \mathbb Z^d \to \mathbb C,\quad h \mapsto Z \left \vert \begin{matrix} h\\ 0 \end{matrix}\right \vert (A;s). $$ The Epstein zeta function $Z$ can be defined via the following Dirichlet series for $s>d$, $$Z \left \vert \begin{matrix} h\\ 0 \end{matrix}\right \vert (A;s)=\sum_{z \in Z^d}' \frac{1}{\big\vert(z+h)^\top A (z+h)\big\vert^{s/2}},\quad s>d,$$ where the primed sum excludes the case $h=-z$, and it is well known since the works of Epstein (Mathematische Annalen 56.4 (1903): 615-644), that the series exhibits a meromorphic continuation to $s\in \mathbb C$, if not all $h\not\in \mathbb Z^d$, and to $\mathbb C\setminus d$ if $h\in \mathbb Z^d$. For $d=1$, we recover the standard Hurwitz zeta function.
The regularity of $Z$ in $s$ has been well established from the start. I am now interested whether someone could point me to a reference where the regularity in $h$ is discussed. Of course, for $s>d$, we find that the function is holomorphic in $h$ for $h\in \mathbb R^d\setminus \mathbb Z^d$, this immediatly follows from the Dirichlet series. However, the situation is less clear for the analytic continuation of Z, thus for $s\le d$. Is Z then still holomorphic in h on $\mathbb R^d\setminus \mathbb Z^d$ with an algebraic pole for $s\in(0,d)$ and a logarithmic singularity for $s=0$?
Thanks to you all!