In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound $$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\ll(\log k)^2.$$ Where might I find a proof of this fact, and can one obtain similar bounds as the level of $f$ varies (a lower bound in particular)? The upper bound, perhaps with a slightly worse exponent, I think can be obtained from the approximate functional equation. The lower bound seems more subtle, however.
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1$\begingroup$ See doi.org/10.2307/2118544, where a proof is written up for Maass forms, and the exact same proof goes through for holomorphic forms (and the authors mention this). $\endgroup$– Peter HumphriesCommented Sep 14, 2020 at 0:55
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1$\begingroup$ As you note the upper bound is a bit inaccurate, and the exponent should be $3$. The lower bound follows from the fact that these $L$-functions have no Siegel zeros, as Peter Humphries points out above. You may also wish to look at Xiannan Li's paper on upper bounds for $L$-values (in IMRN). Upper bounds in general are ok, and a classical zero free region will give lower bounds of that shape. $\endgroup$– LuciaCommented Sep 14, 2020 at 1:47
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