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 $\mathrm{\mid }\zeta (\frac{1}{2}+it){\mathrm{\mid }}^{2k}$ 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 $O(1\mathrm{/}\log T)$ from the zero.