On complex explicit formulae connected with the M\"obius function of an elliptic curve

TL;DR

This paper investigates the analytic continuation and explicit formulae of a complex function related to the L-function of an elliptic curve, revealing its meromorphic nature and functional equation.

Contribution

It provides the first explicit formula for the function $m(z, E)$ in a specific strip and establishes its meromorphic continuation and functional equation.

Findings

$m(z, E)$ extends meromorphically to the entire complex plane.

An explicit formula for $m(z, E)$ is derived in the strip $| ext{Im} z|<2 ext{pi}$.

$m(z, E)$ satisfies a certain functional equation.

Abstract

We study analytic properties function $m(z, E)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $|\Im{z}|<2\pi$.