Siegel's article "The volume of the fundamental domain for some infinite groups": trouble with understanding computations

Question

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed several domains) is clear, I do not quite understand how to prove some statements he makes about his computations, namely, on the page 3, the integral $$\int_{F_{1}}{|X|^{\frac{s}{2}-1} \sum_{|K|=0} e^{-\pi\cdot tr(K'XK)} dX}$$ is proclaimed to be a regular function of $s$ near $s=1$. (Siegel uses very specific notations, $|X|$ is the determinant, and $X'$ is $X$ transposed. $X$ is a positive definite real-valued symmetric matrix $n\cdot n$, $F_{1}$ is the part of the fundamental domain where $det(X)<1$, $K$ is a matrix with integer entries.) I do not see why this is true and, being not very good at studying zeta and theta functions, I would welcome any help. By myself, I tried to change the standard coordinates to the eigenvalues of X and coordinates on corresponding classes of orthogonal matrices, but it was of no use. If there is some profound knowledge concerning theta functions that might explain everything, please share!