Geodesic
Local zeta factors and geometries under $\operatorname{Spec}\mathbf Z$
Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 751-758

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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]).
Keywords: cusp forms, period polynomials, local factors. 
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     author = {Yu. I. Manin},
     title = {Local zeta factors and geometries under $\operatorname{Spec}\mathbf Z$},
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     publisher = {mathdoc},
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     number = {4},
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Yu. I. Manin. Local zeta factors and geometries under $\operatorname{Spec}\mathbf Z$. Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 751-758. http://geodesic.mathdoc.fr/item/IM2_2016_80_4_a6/