Solution
= Solution
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.