The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler.
In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here.
The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GLm.
In his notebooks, Ramanujan generalized the Euler product for the zeta function as for s > 1 where Lis(x) is the polylogarithm.
The Leibniz formula for π can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios (fractions where numerator and denominator differ by 1): where each numerator is a prime number and each denominator is the nearest multiple of 4.