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I am a huge fan of the Riemann Zeta function's functional equation:

$$\large{\color\green{\zeta(x)=2^x \Gamma(1-x)\zeta(1-x)\pi^{x-1}\sin\frac{\pi x}{2}}}$$

I am curious as to what conditions on $x$ there are in order to make this equation true. I am researching powers of $\pi$ in the use of integrals and sums, and wanted to know what exactly $x$ can and cannot be. Perhaps I am misdefining functional equation, but I really can't find any straightforward answer. Thank you.

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Your $x$ can be any complex number. However, yes, this requires extending the definition of $\zeta(x)$ beyond the region where real part of $x$ is $>1$, where it is defined directly by the (convergent) series $\sum_{n\ge 1}1/n^x$.

Yes, there is potentially concern about what it means to raise positive integers, or $\pi$, to complex powers like $n^x$ and $\pi^x$. This is not meant to be a "big deal"... for positive real $r$, $r^x$ means $e^{x\log r}$, where the logarithm is the real one.

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