There are examples that don't come from number theory, although it's not much simpler. Specifically, the Lefschetz zeta function.

Let $X$ be a compact manifold of dimension $d$ and let $f: X\to X$ be a map. Let $L(f^n)$ be the number of fixed points of $f^n$, counted with appropriate multiplicity. Then if we define

$$\zeta_f(t) = e^{ \sum_{n=1}^{\infty} L(f^n) \frac{t^n}{n} }$$

by the Lefschetz fixed point formula we have

$$ \zeta_f(t) =e^{ \sum_{n=1}^{\infty} L(f^n) \frac{t^n}{n} } = e^{ \sum_{n=1}^{\infty} \sum_{i=1}^{d} (-1)^i \operatorname{tr} ( f^n, H^i(X,\mathbb Q))\frac{t^n}{n} }=\prod_{i=1}^{d} \left( e^{ \sum_{n=1}^{\infty} \operatorname{tr} ( f^n, H^i(X,\mathbb Q))\frac{t^n}{n} }\right)^{(-1)^i} $$

$$\prod_{i=1}^{d}\left( \det ( 1 - t f, H^i(X,\mathbb Q)) \right)^{(-1)^i}$$

Now using the fact that the eigenvalues of $f$ on $H^i$ are $\deg f$ divided by the eigenvalues of $f$ on $H^{d-i}$, which comes from Poincare duality, by substituting each term we can prove the functional equation (if $d$ is even)

$$ \zeta_f ((\deg f)^{-1} t^{-1} ) = \prod_{i=1}^{d}\left( \det ( 1 - (\deg f)^{-1} t^{-1} f, H^i(X,\mathbb Q)) \right)^{(-1)^i}= \prod_{i=1}^{d}\left( \det ( 1 - t^{-1} f^{-1} , H^{d-i} (X,\mathbb Q)) \right)^{(-1)^i}= \prod_{i=1}^{d}\left( \det ( 1 - t^{-1} f^{-1} , H^{i} (X,\mathbb Q)) \right)^{(-1)^i}$$

$$=\prod_{i=1}^{d}\left( \det ( tf - 1, H^i(X,\mathbb Q)) \right)^{(-1)^i} t^{ (-1)^{i+1} \dim H^i(X,\mathbb Q)} / \det(f,H^i(X,\mathbb Q)) =\prod_{i=1}^{d}\left( \det ( 1- tf 1, H^i(X,\mathbb Q)) \right)^{(-1)^i} t^{ -\chi(M)} C = C t^{-\chi(M)} \zeta_f(t) $$ for some constant $C$.

The number theory one is proved exactly the same way.