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Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large. Then $$L(-1+it,f)\ll_f \log^c q(f)t$$ holds, for some conatant $c$ ? Where $q(f)$ is the analytic conductor, $|t|>2$.

We ever see the bound like $L(1/2,f)$, i want to know the corresponding bound for $L(s+it)$ with $\Re s<0$. I search the H. Iwaniec and E. Kowalski's book, it seems there are no records. Please, if some references talk about it, show me their names.

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  • $\begingroup$ Use the functional equation and Stirling's formula. $\endgroup$
    – Lucia
    Commented Oct 14, 2013 at 2:41

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This is not true. By standard bounds $|L(2+it,f)|\asymp_f 1$, hence by the functional equation and standard bounds for the gamma function $|L(-1+it,f)|\asymp_f |t|^{3/2}$ for $|t|>1$. Here $A\asymp_f B$ means that $c_1 B<A<c_2 B$ with positive constants $c_1$ and $c_2$ depending on $f$. Note that I normalize $L$-functions to have center $1/2$, as is customary in analytic number theory and in the theory of automorphic forms.

In general, for a principal automorphic $L$-function $L(s,\pi)$ we have the convexity bound $$ L(\sigma+it,\pi) \ll_{\sigma,\epsilon} C(1/2+it,\pi)^{\max(1/2-\sigma,1/2-\sigma/2,0)+\epsilon},\qquad \sigma\in\mathbb{R}, $$ where $C(1/2+it,\pi)$ is the analytic conductor of $L(1/2+it,\pi)$, and this is hard to improve. In particular, this bound is not known in general with the factor $C(1/2+it,\pi)^\epsilon$ replaced by $\log^AC(1/2+it,\pi)$, although it is certainly known in the special case you are considering.

It is known that the Grand Riemann Hypothesis implies the Grand Lindelöf Hypothesis, which would replace $\max(1/2-\sigma,1/2-\sigma/2,0)$ above by $\max(1/2-\sigma,0)$, and the factor $C(1/2+it,\pi)^\epsilon$ by $\log^AC(1/2+it,\pi)$.

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    $\begingroup$ Actually the convexity bound on the $1/2$ line does hold without the $C^{\epsilon}$ factor. This is due to Heath-Brown arxiv.org/abs/0809.1752 , which appeared in Acta Arithmetica. $\endgroup$
    – Lucia
    Commented Oct 14, 2013 at 18:25
  • $\begingroup$ @Lucia: Thanks for this comment. Let us mention that Heath-Brown requires some control on the Langlands parameters at the various primes, but these are known by the work of Luo-Rudnick-Sarnak. The bound I displayed is more general, it relies on the work of Molteni (which in turn needs the Luo-Rudnick-Sarnak bounds if I recall correctly). $\endgroup$
    – GH from MO
    Commented Oct 14, 2013 at 19:02
  • $\begingroup$ Brilliant! Thanks for your reply. In IK'book, $(5.20)$ states $L(\sigma+it,f)\ll_{\epsilon,f}C(\frac{1}{2}+it,f)^{\frac{1-\sigma}{2}+\epsilon}$, when $0\le\sigma\le 1$ and $t$ is large. $C^\epsilon$ factor may be not omitted in the convexity bound above. $\endgroup$
    – H.Flip
    Commented Oct 17, 2013 at 8:04
  • $\begingroup$ @Houfei: The bound I stated is stronger and more general than (5.20) that you quote. First, the implied constant only depends on $\sigma$ (not on $\pi$). In fact this dependence is continuous in $\sigma$. Second, I stated it for all $\sigma$ (not just $0\leq\sigma\leq 1$). $\endgroup$
    – GH from MO
    Commented Oct 20, 2013 at 1:50

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