It is expected that the **Euler product** of an L-function of degree $d$ and conductor $N$ can be written as
$$L(s)=\prod_p L_p(s)$$
where for $p\nmid N$
$$L_p(s)=\prod_{n=1}^d \left( 1-\frac{\alpha_{n}(p)}{p^s}\right)^{-1} \text{ with } |\alpha_{n}(p)|=1$$
and
for $p\mid N$,
$$L_p(s)=\prod_{n=1}^{d_p}\left( 1-\frac{\beta_{n}(p)}{p^s}\right)^{-1} \text{ where } d_p<d \text{ and } |\beta_n(p)|\le 1.$$
The functions $L_p(s)$ are called **Euler factors** (or **local factors**).

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-05-11 16:51:32

**Referred to by:**

- artin.lfunction
- cmf.lfunction
- g2c.good_lfactors
- g2c.lfunction
- intro.tutorial
- lfunction
- lfunction.central_character
- lfunction.critical_strip
- lfunction.dirichlet
- lfunction.gamma_factor
- lfunction.rational
- lfunction.selberg_class.axioms
- mf.gl2.history.varieties
- rcs.rigor.lfunction.curve
- lmfdb/hypergm/templates/hgm_family.html (line 105)
- lmfdb/lfunctions/templates/Lfunction.html (line 238)
- lmfdb/lfunctions/templates/MaassformGL2.html (line 8)
- lmfdb/lfunctions/templates/ellipticcurve.html (line 8)
- lmfdb/lfunctions/templates/ellipticcurve.html (line 24)
- lmfdb/lfunctions/templates/genus2curve.html (line 7)

**History:**(expand/hide all)

- 2019-05-11 16:51:32 by Andrew Sutherland (Reviewed)
- 2019-04-30 09:54:12 by Stephan Ehlen (Reviewed)
- 2019-04-30 09:02:17 by Stephan Ehlen (Reviewed)
- 2012-06-29 09:20:22 by David Farmer

**Differences**(show/hide)