Cfrgtkky

Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(fotimes g,s)=zeta(2s)sum_{ngeq1}frac{lambda_f(n)lambda_g(n)}{n^s}$ be the Rankin-Selberg L-Function where $lambda_f$ and $lambda_g$ are the respective normalized Hecke-Eigenvalues.

I'm looking for a simple upper bound of $L(fotimes g,frac12+it)$ in the level aspect which holds under general assumptions as given above. In nearly every piece of literature on subconvexity bounds (e.g. Introduction of "Rankin Selberg L-functions in the level aspect" by Kowalski, Michel, Vanderkam) there is the hint that the convexity bound ($sqrt N$ in the level aspect, $k$ in the weight aspect) is achieved by applying the Phragmén-Lindelöf principle. As far as I understood, this requires another bound of the function on the edges of the vertical strip $0<Re(s)<1$. So my questions are

• Is there an elementary proof to this problem?


• What bound can I use for $L(fotimes g, 1)$?


• How do I take care of the residue in $s=1$ for $f=overline{g}$?

To provide a brief answer which can be found in "Analytic Number Theory" of Iwaniec and Kowalsky:

We use the absolute convergence of $L(f,s)$ in $sigma>1$ (or the bound $|lambda(n)|leq tau(n)$), the functional equation, Stirling's estimate for the gamma function and the Phragmen-Lindelöf convexity principle.

All mentioned formulas can be found in the book as well.