Local zeta factors and geometries under Spec Z

The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda--structure of the polynomial ring as Borger's descent data to $\bold{F}_1$ and suggests its role in possible relation of $Γ_{\bold{R}}$--factor to "real geometry over $\bold{F}_1$" (cf. also [CoCons2]).

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