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