  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 $|\Im{z}|<2\pi$.
