In what sets are these functions analytic? $$\begin{align} 1. \qquad & \int_0^z \exp(\zeta^2)\,d\zeta \\ 2.\qquad & \int_0^\infty t^{z-1}\exp(-t)\, dt\\ 3. \qquad& \sum_{n=1}^ \infty n^{-z} \end{align}$$
For number 2 I think it is analytic at all of the complex numbers except the integers, but that is just random guess. No idea for number 1. Any suggestion would help. Thanks.