Consider the classical real analytic Eisenstein series $$ E(z,s)=\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ 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.

(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 $s$, we have $E(z,s)=O(y^{\sigma})$ as $y\rightarrow \infty$ where $\sigma=\max(\Re(s),1-\Re(s))$.

Of course if $C\in\mathbb{C}$, then $E(z,s)+C$ satisfies (1), (2), (3), (4) and (5).

Q1 Do the properties (1), (2), (3), (4) and (5) characterize $E(z,s)$ up to a constant ?

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)