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 $zeta_{Lambda}(s) = sumlimits_{n=0}^{infty} a_n n^{-s}$, 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.