  Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery \cite{GGM}
studied the second moment of the log-derivative of $\zeta$, shifted away from
the half line by $a/\log T$, and its connection with the pair correlation
conjecture. In this paper, we consider a weighted version of this problem,
where the average is tilted by $|\zeta(\frac{1}{2}+it)|^2$. More precisely, we
provide an upper and a lower bound for the second moment of zeta times its
logarithmic derivative, $a/\log T$ away from the critical line.
