Horizontal factorizations of certain Hasse-Weil zeta functions -- a remark on a paper of Taniyama (Q1947823)

An easy calculation (see, for instance, \textit{K. Joshi} and \textit{C. S. Yogananda} [Acta Arith. 91, No. 4, 325--327 (1999; Zbl 0959.11037)]) shows that \[ \zeta(s- 1)\zeta(s)^{-1}= \prod^\infty_{n=1}\, \prod_{\chi\in X(n)} L(\xi,s)\quad\text{for Re\,} s> 2, \] where \(X(n)\) stands for the group of the Dirichlet characters modulo \(n\). The left-hand side of this identity may be regarded as the Hasse-Weil zeta function of the scheme \(G_m:= \text{Spec\,}\mathbb Z[t, t^{-1}]\). In the work of \textit{Y. Taniyama} [J. Math. Soc. Japan, 9, 330--366 (1957; Zbl 0213.22803)] analogous formulae had been obtained for the Hasse-Weil zeta functions of Abelian schemes. The authors prove such formulae for extensions of Abelian schemes by a torus and discuss some further generalizations.