Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

The zeta function of an integral lattice $\Lambda$ is the generating function $\zeta_{\Lambda}(s) = \sum\limits_{n=0}^{\infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $\Lambda$ of index $n$. We derive a formula for the zeta function of $\Lambda_1 \otimes \Lambda_2$, where $\Lambda_1$ and $\Lambda_2$ are $\mathbb{Z}$-orders contained in finite-dimensional semisimple $\mathbb{Q}$-algebras that satisfy a "locally coprime" condition. We apply the formula obtained above to $\mathbb{Z}S \otimes \mathbb{Z}T$ and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes $(X,S)$ and $(Y,T)$ in several cases where the $\mathbb{Z}$-orders $\mathbb{Z}S$ and $\mathbb{Z}T$ are locally coprime and their zeta functions are known.