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<h1 align=center> Euler product
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An {\it 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 \begin{rawhtml}<a href="../4a">Riemann $\zeta$-function</a>\end{rawhtml} has Euler product
$$\zeta(s) = \prod_p \left(1-p^{-s}\right)^{-1}.
$$

A \begin{rawhtml}<a href="../9a">Dirichlet $L$-function</a>\end{rawhtml} has Euler product
$$L(s,\chi) = \prod_p \left(1-\chi(p) p^{-s}\right)^{-1}.
$$

The \begin{rawhtml}<a href="../66a">Dedekind zeta function</a>\end{rawhtml} of a number field $K$
has Euler product
$$\zeta_K(s) = \prod_{\mathfrak p} \left(1-N{\mathfrak p}^{-s}\right)^{-1},
$$
where the product is over the prime ideals of ${\mathcal O}_K$.

An \begin{rawhtml}<a href="../13a">$L$-functions associated with a newform</a>\end{rawhtml} 
$f\in S_k(\Gamma_0(N))$ or a 
\begin{rawhtml}<a href="../14a">Maass newform</a>\end{rawhtml} 
$f(z)$ on $\Gamma_0(N)$
has Euler product
$$L(s,f) = \prod_{p|N} \left(1-a_p p^{-s}\right)^{-1}
       \prod_{p\nmid N}
        \left(1-a_p p^{-s}+ \chi(p) p^{-2s+1}\right)^{-1} .
$$

\begin{rawhtml}<a href="../44a">$GL(r)$ $L$-functions</a>\end{rawhtml} have Euler products where
almost all of the local factors are (reciprocals of) polynomials in
$p^{-s}$ of degree~$r$.

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<p> <br>
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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