Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a function on $(z,s)\in\mathfrak{h}\times\mathbf{C}$. The function $E(z,s)$ satisfies the following properties
(1) For a fixed $z\in\mathfrak{h}$, $s\mapsto E(z,s)$ is holomorphic except with poles of order $1$ at $s=1$ and $s=0$ with residues $1/2$ and $-1/2$ respectively (the knowledge of one residue implies the knowledge of the other from the functional equation in $s$, see below).
(2) $E(z,s)$ is $SL_2(\mathbb{Z})$-invariant in $z$
(3) $\Delta_h E(z,s)=s(1-s)E(z,s)$ where $\Delta_h$ is the hyperbolic Laplacian.
(4) $E(z,s)=E(z,1-s)$
(5) For a fixed $s\in\mathbf{C}\backslash\{\frac{1}{2}\}$, we have $E(z,s)=O(y^{\sigma})$ as $y\rightarrow \infty$ where $\sigma=\max(\Re(s),1-\Re(s))$.
Q1 Do the properties (1), (2), (3), (4) and (5) characterize $E(z,s)$ ?
Q2 Is there some redundancy among properties (1), (2), (3), (4) and (5)?
Q3 What is a good way to characterize what $E(z,s)$ is ? (I guess that representation theorists should have something nice to say for Q3)
added Note that $E(z,\frac{1}{2})$ is not square integrable. Indeed,
looking at the constant term of the Fourier series in $z$ of $E(z,1/2)$ we find
that $E(z,1/2)\sim Cte\cdot\log(y)\sqrt{y}$. So if one integrates in the usual fundamental domain $\mathcal{D}_{T}$ of $SL_2(\mathbb{Z})$, up to height $T$, with respect to the Poincare volume, we find that
$$
\int_{\mathcal{D}_T}|E(z,1/2)|^2\frac{dxdy}{y^2}\sim \int_{1}^{T} \frac{\log(y)^{2}dy}{y}\sim \frac{1}{3}\log(T)^3.
$$
So as $T\rightarrow \infty$ the integral diverges. Note though that it is
"almost" square integrable in the sense that it diverges extremely slowly.
Best Answer
For $Re(s)>1$, the function $E(z,s)$ is smooth on $\mathbb{H}$ and satisfies $(2)$, $(3)$ and $$ (*) \ E(z,s)-\xi(2s) \cdot y^s=O(y^{1-s}) \ \text { as } y \rightarrow +\infty. $$ (Here $\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed Riemann zeta function.) One can prove this by computing the Fourier expansion of $E(z,s)$. These properties characterize $E(z,s)$ since the difference of any two functions satisfying them is square-integrable on $SL_2(\mathbb{Z})\backslash \mathbb{H}$ (think of the usual fundamental domain) and every eigenvalue $\lambda$ of $\Delta$ in $L^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$ is $\lambda \geq 0$.
Together with having meromorphic continuation to $s \in \mathbb{C}$ for fixed $z$, this characterises $E(z,s)$.
(Of course, this sort of characterisation of $E(z,s)$ is well-known: see e.g. Lemma 2.5.1 in these notes or Lemma 1 in here.)