Local zeta factors and geometries under $\operatorname{Spec}\mathbf Z$

The first part of this note shows that the odd-period polynomial of each Hecke cusp eigenform for the full modular group produces via the Rodriguez-Villegas transform ([1]) a polynomial satisfying the functional equation of zeta type and having non-trivial zeros only in the middle line of its critical strip. The second part discusses the Chebyshev lambda-structure of the polynomial ring as Borger's descent data to $\mathbf{F}_1$ and suggests its role in a possible relation of the $\Gamma_{\mathbf{R}}$-factor to `real geometry over $\mathbf{F}_1$' (cf. [2]).