I would like to give a presentation of Tate's proof of the functional equation, emphasizing the parallel between this proof and one of Riemann's. The aim is mainly to introduce and motivate the adelic setting as a powerful tool to do analytic number theory.
However, since the talk would be intended to non-specialists, I would like to motivate the importance of the classical functional equation, without having to sink in deep number theory, which will lost everyone. So here is my question:
What are the appealing and elementary motivations or applications to the $\zeta$ functional equation?