# Convexity Bound of Rankin-Selberg L-Function

by Nodt Greenish   Last Updated November 13, 2018 18:20 PM

Let $$f,g$$ be primitive modularforms of arbitrary levels $$N_1,N_2$$, trivial nebentypus and same weight $$k$$. Let $$L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$$ 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 $$L(f\otimes g,\frac12+it)$$ 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 $$L(f\otimes g, 1)$$?
• How do I take care of the residue in $$s=1$$ for $$f=\overline{g}$$?
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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.

Nodt Greenish
November 13, 2018 18:17 PM