Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.


I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ is absolutely convergent in $\{s\in\mathbb{C}:\operatorname{Re}(s)>1\}$. I can only convince myself for the special case that $\pi$ and $\pi'$ are contragredient.


Why is this true for other cases?



Absolute convergence of Rankin–Selberg series
This is an elaboration of Lucia's [comment](https://mathoverflow.net/questions/169251/absolute-convergence-of-rankin-selberg-series#comment428677_169251). Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following the proof of Proposition 6 in Molteni's paper [Upper and lower bounds at $s = 1$ for certain Dirichlet series with Euler product](https://doi.org/10.1215/S0012-7094-02-11114-4) (Duke Math. J. 111 (2002), 133-158), we see that there exists a polynomial $P\_k\in\mathbb{N}[x\_1,\dots,x\_k]$ such that
\begin{gather\*}
k!\ a\_{\pi\times\pi'}(p^k) = P\_k(\sigma\_1\tau\_1,\dotsc,\sigma\_k\tau\_k), \\
k!\ a\_{\pi\times\tilde\pi}(p^k) = P\_k(\lvert\sigma\_1\rvert^2,\dotsc,\lvert\sigma\_k\rvert^2), \\
k!\ a\_{\pi'\times\tilde\pi'}(p^k) = P\_k(\lvert\tau\_1\rvert^2,\dotsc,\lvert\tau\_k\rvert^2),
\end{gather\*}
where $\sigma\_h$ (resp. $\tau\_h$) denotes the $h$-th power sum of the Langlands parameters of $\pi$ (resp. $\pi'$) at $p$. Using Cauchy–Schwarz, it follows that
$$ \lvert a\_{\pi\times\pi'}(p^k)\rvert\leq \sqrt{a\_{\pi\times\tilde\pi}(p^k)a\_{\pi'\times\tilde\pi'}(p^k)}.$$
Note that the coefficients on the right are nonnegative. Hence for all positive integers $n$, we have
$$ \lvert a\_{\pi\times\pi'}(n)\rvert\leq \sqrt{a\_{\pi\times\tilde\pi}(n)a\_{\pi'\times\tilde\pi'}(n)}\leq (a\_{\pi\times\tilde\pi}(n)+a\_{\pi'\times\tilde\pi'}(n))/2.$$
It follows that the absolute Dirichlet series for $L(s,\pi\times\pi')$ is majorized by the average of the Dirichlet series for $L(\sigma,\pi\times\tilde\pi)$ and $L(\sigma,\pi'\times\tilde\pi')$. The latter converges for $\sigma>1$ by Landau's lemma, hence $L(s,\pi\times\pi')$ converges absolutely for $\Re(s)=\sigma>1$.


**Added 1.** The above argument requires that $p$ is an unramified prime for $\pi$ and $\pi'$, hence also that $n$ is only divisible by such primes. This does not alter the final conclusion though, because omitting or including finitely many Euler factors does not alter absolute convergence of the corresponding Dirichlet series in the half-plane $\Re(s)>1$.


**Added 2.** For a related and more complete result concerning the logarithmic derivative of Rankin–Selberg $L$-functions, see Brumley's Appendix to the exciting new preprint [Weak subconvexity without a Ramanujan hypothesis](https://arxiv.org/abs/1804.03654) of Soundararajan and Thorner.


**Added 3.** The above bounds have been extended to all prime powers $p^k$ and all positive integers $n$. See Lemma 3.1 in the nice paper [Exponential sums with multiplicative coefficients without the Ramanujan conjecture](https://doi.org/10.1007/s00208-020-02108-z) of Jiang–Lü–Wang. The proof relies on Brumley's Appendix that I emphasized in the previous section.


