A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by \(|\zeta (\frac{1}{2}+it)|^{2k}\) for \(k=1\) and, for test functions with Fourier support in \((-\frac{1}{2},\frac{1}{2})\), for \(k=2\). As a consequence, for \(k=1,2\), we deduce under the Riemann hypothesis that \(T(\log T)^{1-k^2+o(1)}\) non-trivial zeros of \(\zeta \), of imaginary parts up to T, are such that \(\zeta \) attains a value of size \((\log T)^{k+o(1)}\) at a point which is within \(O(1/\log T)\) from the zero.