In fact, for all complex "k" (in your notation), that series (analytically continued, anyway, because it only converges for Re(1/2+ik)>1...) is an eigenfunction for the hyperbolic Laplacian. These eigenfunctions are never square-integrable on the quotient $SL(2,\mathbb Z)\backslash H$, however. Nevertheless, integrals (often called "wave packets") over $k$ of these Eisenstein series are square integrable.
Apart from the periodicity requirements, there really are no boundary conditions here. We have a continuum of these "Eisenstein series".
In fact, the bulk of the space of square-integrable $SL(2,\mathbb Z)$-periodic functions on the upper half-plane is made of up "cuspforms", which are genuine square-integrable eigenfunctions for this Laplace-Beltrami operator. It is very difficult to give explicit numerical examples of these cuspforms, despite the fact (proven by Selberg in the 1950s, and thereafter) that they satisfy "Weyl's Law" about asymptotics of eigenfunctions.