Extreme values of L-functions

A celebrated theorem of Selberg indicates that the typical size of
$|\zeta(\frac{1}{2}+it)|$ is $\exp( \sqrt{\frac{1}{2}\log \log
T})$. What about extreme values of  $\zeta(\frac{1}{2}+it)$? If
we choose $t$ randomly from $[0, T]$, then it has been shown recently
that $\max_{ t\in[0,T]} | \zeta( \frac{1}{2}+it)| \gg \exp (
\sqrt{\frac{2\log T\log\log \log T}{\log \log T}})$. What if we
choose $t$ at some discrete points? In this talk, we are interested in
choosing $t$ to be the imaginary part of the zeros of some other
$L$-functions on the critical line. In particular, under GRH, we can
show that $\max_{L(\rho,\chi)=0, T\leq Im \rho\leq 2T} | \zeta( \rho)| \gg \exp (
c\sqrt{\frac{\log T}{\log \log T}}), c>0$, where $L(s, \chi)$ is a Dirichlet $L$-function.