In general, a Dirichlet series of the form
1 + a(p)p-s + a(p2)p-2s + ... .
In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that a(n) be multiplicative: this says exactly that a(n) is the product of the a(pk) when n factorises as the product of the powers pk of distinct primes p.
In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region Re(s) > C: that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.
P(p,s) = 1/(1 - a(p)p-s)
as is the case for the Riemann zeta-function (with a(n) = 1), and more generally for Dirichlet characters. 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.