Functional equations for zeta functions of $\mathbb{F}_1$-schemes

For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $ζ(s)=\prod(s-i)^{-a_i}$. Define $χ=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\mathbb{F}_1$-zeta function satisfies the functional equation $ζ(n-s) = (-1)^χζ(s)$. We further show that the $\mathbb{F}_1$-zeta function $ζ(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $ζ(r+N-s) = (-1)^χ( ζ(s) )^{(-1)^r}$.

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