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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?

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    This is not an answer, but I believe that one of the obvious applications of the functional equation is that it provides a full analytic continuation of $\sum_{n=1}^{\infty}\frac{1}{n^s}$ into the domain $\Re(s) <1$. The 'appealing' side of this continuation is beautifully visualised in this video (start e.g. at 10:00): https://www.youtube.com/watch?v=sD0NjbwqlYw – 2017-01-14
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    @Agno Thanks for the video, however I was actually trying to avoid this kind of *per se* motivation, which will not resist to skeptics who does not are used to the ubiquitous zeta function. Do you then deduce from the analytic continuation some good and accessible properties of prime numbers? – 2017-01-14
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    I am not an expert on this, but my logic would be that the analytic continuation allows you to find both the trivial (derived directly from the functional equation) and the non-trivial zeros of $\zeta(s)$. Each zero then encodes information about the distribution of the primes. The infinite summation of all these zeros (expressed as spiralling waves) can then be used to generate the various prime-counting functions (e.g. $\pi(x)$ or $\Psi(x)$). – 2017-01-14
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    My knowledge about the Riemann Zeta is very poor. Any case I can add this curiosity, for example first paragraph of Tao, *Tate’s proof of the functional equation*, What's the new? (2008), where $\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$. Since we start with analyticity on $\Re s>1$ and $\zeta(s)$ has an Euler product then when you discover the symmetry of $\xi(s)$ you can can define $\Gamma_\infty(s)$ or in words of Garrett, *Riemann’s Explicit/Exact formula 24 sept. 2015*, see in his home page, Remark 2.2.7 **is the Euler factor corresponding to the prime** $\infty$. – 2017-01-15

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