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

The zeta function of an integral lattice $\mathrm{\Lambda }$ is the generating function ${\zeta }_{\mathrm{\Lambda }}(s)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{n}^{-s}$, whose coefficients count the number of left ideals of $\mathrm{\Lambda }$ of index $n$. We derive a formula for the zeta function of ${\mathrm{\Lambda }}_1\otimes {\mathrm{\Lambda }}_2$, where ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ 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.