A trivial and artificial example is g(z), defined as the sum of G(n, z) for n from 1 to infinity, where G(n, z) = 0 except if z is -1, in which case it is n.

Thus g(-1) is 1 + 2 + 3 + ... while g(z) is otherwise 0 + 0 + 0 + ... = 0. And then the zero function is the unique analytic continuation of g.

The reason people care about the Riemann zeta function is because of its deep connections to analysis, number theory, and physics.