Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture? 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.
 A: 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 $\log^AC(1/2+it,\pi)$.
