Zeros of $L$-functions in families near the critical line

We combine the relative trace formula with analytic methods to obtain zero density estimate for $L$-functions in various families of automorphic representations for $\mathrm{G}\mathrm{L}(m)$. Applications include strong bounds for the average analytic rank of these $L$-functions at the central point and average equidistribution results for the imaginary parts of the zeros.