Asymmetry of a class of Mellin transforms

We consider a class of Mellin transforms $\mu_\eta$ associated with a class of functions $\eta$, defined for every complex $s$ in the critical strip. We establish sufficient conditions on $\eta$ under which $\mu_\eta(s)$ and $\mu_\eta(1-s)$ cannot both vanish outside the critical line for sufficiently large imaginary parts of $s$. An application is given to the case in which $\eta$ is the fractional part function, for which the zeros of $\mu_\eta$ coincide with the zeros of the Riemann zeta function.