Abstract
For a scheme X whose F q -rational points are counted by a polynomial N ( q ) = ∑ a i q i , the F 1 -zeta function is defined as ζ X ( s ) = ∏ ( s − i ) − a i . Define χ = N ( 1 ) . In this paper we show that if X is a smooth projective scheme, then its F 1 -zeta function satisfies the functional equation ζ X ( n − s ) = ( − 1 ) χ ζ X ( s ) . We further show that the F 1 -zeta function ζ G ( s ) of a split reductive group scheme G of rank r with N positive roots satisfies the functional equation ζ G ( r + N − s ) = ( − 1 ) χ ( ζ G ( s ) ) ( − 1 ) r .
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