Quite generally I thought that every holomorphic function defined in some domain can be analytically continued into the entire complex plane. However the dedekind eta function seems to not have such an analytic continuation to the lower half as is stated here Dedekind Eta Function.
So what is the problem here precisely? As an example the zeta function defined via the dirichlet series is only convergent for s>1. Probably it also converges for s=1 if Im(s) is not equal zero. Still it can be continued.
Now the dedekind eta (or similar functions as the jacobi theta) are only defined for Im(t)>0. Again I would assume it to converge on the real axis if t is not an integer. So what prevents it of being continued?