We give a complete characterization of the solutions $F(s)$ of the analog in the Selberg class of Hecke's functional equation of conductor 5, namely [ left(rac{sqrt{5}}{2pi} ight)^s Gamma(s+mu) F(s) = omega left(rac{sqrt{5}}{2pi} ight)^{1-s} Gamma(1-s+overline{mu}) overline{F(1-overline{s})} ] with $Re{mu}geq 0$ and $|omega|=1$. The proof is based on several results from our theory of nonlinear twists of $L$-functions, applied to obtain a full description of the Euler factor of $F(s)$ at $p=2$, and then on some ideas from a 1995 paper by J.B.Conrey and D.W.Farmer on converse theorems for Euler products.
On a Hecke-type functional equation with conductor q=5
A. PERELLI
2018
Abstract
We give a complete characterization of the solutions $F(s)$ of the analog in the Selberg class of Hecke's functional equation of conductor 5, namely [ left(rac{sqrt{5}}{2pi} ight)^s Gamma(s+mu) F(s) = omega left(rac{sqrt{5}}{2pi} ight)^{1-s} Gamma(1-s+overline{mu}) overline{F(1-overline{s})} ] with $Re{mu}geq 0$ and $|omega|=1$. The proof is based on several results from our theory of nonlinear twists of $L$-functions, applied to obtain a full description of the Euler factor of $F(s)$ at $p=2$, and then on some ideas from a 1995 paper by J.B.Conrey and D.W.Farmer on converse theorems for Euler products.
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