Euler product

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

The Riemann $\zeta$-function has Euler product

\begin{displaymath}\zeta(s) = \prod_p \left(1-p^{-s}\right)^{-1}.

A Dirichlet $L$-function has Euler product

\begin{displaymath}L(s,\chi) = \prod_p \left(1-\chi(p) p^{-s}\right)^{-1}.

The Dedekind zeta function of a number field $K$ has Euler product

\begin{displaymath}\zeta_K(s) = \prod_{\mathfrak p} \left(1-N{\mathfrak p}^{-s}\right)^{-1},

where the product is over the prime ideals of ${\mathcal O}_K$.

An $L$-functions associated with a newform $f\in S_k(\Gamma_0(N))$ or a Maass newform $f(z)$ on $\Gamma_0(N)$ has Euler product

\begin{displaymath}L(s,f) = \prod_{p\vert N} \left(1-a_p p^{-s}\right)^{-1}
...p\nmid N}
\left(1-a_p p^{-s}+ \chi(p) p^{-2s+1}\right)^{-1} .

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

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