On certain zeta integral: Transformation formula

We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $\zeta$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $\zeta$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $\zeta$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $\zeta$ when restricted to points fixed by elements of $G$. As an illustration of this principle, we evaluate certain generalized Lambert series at roots of unity, establishing pertinent algebraicity results. Also, we express the Barnes' multiple gamma function at roots of unity as a certain infinite product. It should be mentioned that this work also considers twisted versions of $\zeta$.