I was just solving an Exercise, where we looked at an analytic function $\phi : \mathbf{H} \to \mathbf{C}$, which is automorphic and $\phi (z) = \mathcal{O}(y^{-C})$ for all $C > 0$ as $z \to i \infty$ (we write $z = x + iy$). I think that this implies that $\phi$ is constant, since it is a modular form of weight zero.
The solution to the Exercise however is three pages long and is about proving that $$\Lambda_\phi (s) := 2 \pi^{-s} \Gamma (s) \zeta(s) \mathcal{M}(\phi)(s - 1)$$ satisfies the functional equation $$\Lambda_\phi (s) = \Lambda_\phi(1 - s)$$ where $\mathcal{M}(\phi)(s)$ is the Mellin-transform $\int_0^\infty \phi(x + iy) y^{s - 1} dy$.
This is referred to as "the simplest case of the Rankin-Selberg-method". The statement gets rather easy if $\phi$ is constant. So my question is: What was the actual point of the Exercise? I was hoping for someone to recognize a formulation of the Rankin-Selberg-method (which I have only seen stated differently in the sources I considered) or of a general tool that is used in the theory.
Thanks!
Added: Thinking about the problem again and looking at the specific example of the non-holomorphic Eisenstein series $$E(z, s) = \sum_{(m, n) \in \mathbf{Z}^2, \, (m, n ) \neq (0, 0)} \frac{y^s}{\vert m z + n \vert^{2s}}$$ I suppose that the only weakening on $\phi$ that is necessary is to make $\phi$ a smooth function in $x$ and $y$, instead of a holomorphic function. But my question is still: "What is the context of this exercise?"