Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f \times f\right) X^{it} dt, $$ where $L(s, f\times f) = \sum a(n)^2 n^{-s}$ is the Rankin-Selberg convolution of a holomorphic $GL_2$ cusp form $f$ with itself and where I intend the normalization to be such that $\tfrac{1}{2}$ is the center of its critical strip; and $X$ is a parameter that will be getting very large.
Assuming the Lindelöf hypothesis, we expect very little contribution from the $L$-function, yielding a hypothetical $T^{1 + \epsilon}$ bound without incorporating the spinning from the $X_{it}$ factor. In fact, we could Cauchy-Schwarz the integral to make this heuristic more formal, but this transforms the problem into the much harder problem of getting the second moment of the Rankin-Selberg $L$-function.
I suspect that it may be possible to get a good bound on this integral without Lindelöf if one took proper care of the oscillation. I'm unaccustomed to the $X^{it}$ factor, and I have not seen moments of this form done before. I'm hoping for references for this integral, or perhaps similar integral moments such as $$ \int_0^T L\left(\tfrac{1}{2} + it, f\right) X^{it} dt $$ or $$ \int_0^T \zeta\left(\tfrac{1}{2} + it\right) X^{it} dt, $$ so that I may get some idea as to how to approach this oscillatory integral moment.
