Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$. Do we know any case that \[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\] holds unconditionally?
I know the conjectured asymptotic formula is $T \log^* T$.
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6$\begingroup$ Yes, this kind of lower bound in $t$-aspect is trivial for any $L$-function. The point is that you can approximate $L(\frac 12+it)$ by some long Dirichlet polynomial $\sum_{n\le T^{r}} a(n)n^{-1/2-it}$, say, with $a(1)=1$, and the $L$-function coming from $GL(r)$, say. If you now integrate with smooth weights $L(1/2+it)$ (without absolute values), then note that only the term $n=1$ contributes. The rest of the terms cancel out and are negligible for smooth weights (rapid decay of Fourier transforms). So the bound $\gg T$ follows. $\endgroup$– LuciaCommented Oct 20, 2014 at 2:22
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see http://www.ams.org/journals/proc/2009-137-11/S0002-9939-09-10012-6/S0002-9939-09-10012-6.pdf ,where the low bound $T$ was obtained unconditionally for any $G_m(\mathbb{Q_A})$ and any power of positive real $k$.
