Compatibility of special value conjectures with the functional equation of zeta functions

Abstract

We prove that the special value conjecture for the Zeta function $\zeta (\mathcal{X},s)$ of a proper, regular arithmetic scheme $\mathcal{X}$ that we formulated in [M. Flach and B. Morin, Doc. Math. 23, 1425–1560 (2018; Zbl 1404.14024)] is compatible with the functional equation of $\zeta (\mathcal{X},s)$ provided that the rational factor $C(\mathcal{X},n)$ we were not able to compute previously has the simple explicit form given in the introduction below.