The second moment of Dirichlet twists of a $$\text {GL}_{4}$$ automorphic L-function

Abstract

Abstract In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic L -function $$L(s, \pi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> on the critical line averaged over characters and conductors, where $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> denotes an irreducible tempered cuspidal automorphic representation of $${\textrm{GL}}_{4}({\mathbb {A}}_{\mathbb {Q}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mn>4</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.

Locations

• Mathematische Zeitschrift - View - PDF