Omega results for twists

We write $f(x)=\Omega(g(x))$ to mean $\limsup_{x\to\infty} f(x)/g(x) >0$.

Hoffstein and Lockhart [ MR 2000j:11071] prove an $\Omega$-result for quadratic twists of the $L$-function associated to a holomorphic newform on $\Gamma_0(N)$:

\begin{displaymath}L(\frac12 ,f,\chi\sb d)=
\Omega(\exp(c\sqrt{\log d}/\log\log d))

for squarefree $d\to\infty$, for some $c>0$.

It would be interesting to prove $\Omega$-results for the collection of all Dirichlet $L$-functions $L(\frac12 , \chi)$, $\chi$ mod $d$, and for twists by all characters of $L(\frac12 , f, \chi)$, for $f$ a cusp form.

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