An (analytic) **L-function** is a Dirichlet series that has an Euler product and satisfies a certain type of functional equation.

It is expected that all L-functions satisfy the Riemann Hypothesis, that all of the zeros in the critical strip are on the critical line. Selberg has defined a class $\mathcal S$ of Dirichlet series that satisfy the Selberg axioms. It is conjectured (but far from proven) that $\mathcal S$ is precisely the set of all L-functions. Selberg's axioms have not been verified for all of the L-functions in this database but are known to hold for many of them.

It is also conjectured that a precise form of the functional equation holds for every element of $\mathcal S$. Under this assumption the functional equation is determined by a quadruple known as the Selberg data, consisting of the degree, conductor, spectral parameters, and sign.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2019-05-19 08:43:41

**Referred to by:**

- artin.lfunction
- cmf.embedding
- cmf.lfunction
- g2c.analytic_rank
- g2c.analytic_sha
- g2c.bsd_invariants
- g2c.good_lfactors
- g2c.hasse_weil_conjecture
- intro.tutorial
- lfunction.analytic_rank
- lfunction.arithmetic
- lfunction.central_point
- lfunction.central_value
- lfunction.coefficient_field
- lfunction.completed
- lfunction.conductor
- lfunction.critical_line
- lfunction.degree
- lfunction.dual
- lfunction.euler_product
- lfunction.functional_equation
- lfunction.invariants
- lfunction.known_degree1
- lfunction.known_degree2
- lfunction.known_degree3
- lfunction.known_degree4
- lfunction.motivic_weight
- lfunction.rh
- lfunction.root_number
- lfunction.selbergdata
- lfunction.spectral_parameters
- lfunction.zeros
- rcs.cande.lfunction
- lmfdb/lfunctions/templates/Degree1.html (line 5)
- lmfdb/lfunctions/templates/Degree2.html (line 5)
- lmfdb/lfunctions/templates/Degree3.html (line 5)
- lmfdb/lfunctions/templates/Degree4.html (line 5)
- lmfdb/lfunctions/templates/LfunctionNavigate.html (line 4)
- lmfdb/lfunctions/templates/LfunctionNavigate.html (line 19)

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

- 2019-05-19 08:43:41 by Andrew Sutherland (Reviewed)
- 2019-05-18 09:08:12 by Andrew Sutherland (Reviewed)
- 2019-05-05 12:58:59 by Andrew Sutherland (Reviewed)
- 2019-05-05 12:55:26 by Andrew Sutherland
- 2019-05-05 12:44:37 by Andrew Sutherland
- 2019-05-05 12:10:09 by Andrew Sutherland
- 2019-04-30 08:41:21 by Stephan Ehlen (Reviewed)
- 2015-09-16 20:12:07 by Christelle Vincent
- 2013-09-05 17:42:05 by David Farmer (Reviewed)

**Differences**(show/hide)