Zeros of 𝐿-functions in families near the critical line

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