Euler product

An Euler product is a representation of an -function as a convergent infinite product over the primes , where each factor (called the local factor at '') is a Dirichlet series supported only at the powers of .

The Riemann $\zeta$-function has Euler product

A Dirichlet $L$-function has Euler product

The Dedekind zeta function of a number field has Euler product

where the product is over the prime ideals of .

An $L$-functions associated with a newform or a Maass newform on has Euler product

$GL(r)$ $L$-functions have Euler products where almost all of the local factors are (reciprocals of) polynomials in of degree .

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