Write $\sigma=\operatorname{Re}s$. The assumed coefficient bounds give
$$
|a_n\overline{b_n}n^{-s}|
\leq C^2n^{(k+l)/2-\sigma}.
$$
The comparison <p-series> converges exactly when
$$
\frac{k+l}{2}-\sigma<-1,
$$
or $\operatorname{Re}s>1+(k+l)/2$. Therefore the <Rankin–Selberg convolution> $L(f,g,s)$ converges absolutely in the stated half-plane.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / c / Solution

{c}
{title2=$L(f,g,s)$}

For cusp forms $f=\sum a_nq^n$ and $g=\sum b_nq^n$, their Rankin–Selberg convolution in the elementary normalization is $L(f,g,s)=\sum_{n\geq1}a_n\overline{b_n}n^{-s}$.


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