All known analytic L-functions have a **functional equation** that can be written in the form
\[
\Lambda(s) := N^{s/2}
\prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
\cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
\]
where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
or half-integer,
\[
\sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
\]
and $\varepsilon$ is the sign of the functional equation.
With those restrictions on the spectral parameters, the
data in the functional equation is specified uniquely. The integer $d = J + 2 K$
is the degree of the L-function. The integer $N$ is the conductor (or level)
of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters
in the functional equation can be used to make up the 4-tuple called the Selberg data.

The axioms of the Selberg class are less restrictive than given above.

Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2020-05-04 09:42:44

**Referred to by:**

- artin.root_number
- cmf.lfunction
- ec.q.analytic_rank
- g2c.analytic_rank
- g2c.hasse_weil_conjecture
- intro.tutorial
- lfunction
- lfunction.central_point
- lfunction.completed
- lfunction.conductor
- lfunction.critical_line
- lfunction.critical_strip
- lfunction.degree
- lfunction.dirichlet
- lfunction.gamma_factor
- lfunction.label
- lfunction.normalization
- lfunction.root_number
- lfunction.selberg_class.axioms
- lfunction.selbergdata
- lfunction.sign
- lfunction.spectral_label
- lfunction.spectral_parameters
- lfunction.trivial_zero
- lfunction.zfunction
- mf.bianchi.sign
- mf.gl2.history.varieties
- rcs.rigor.g2c
- rcs.rigor.lfunction.curve
- lmfdb/lfunctions/templates/Degree1.html (line 7)
- lmfdb/lfunctions/templates/Lfunction.html (line 64)
- lmfdb/lfunctions/templates/LfunctionSearchResults.html (lines 18-19)
- lmfdb/lfunctions/templates/MaassformGL2.html (line 14)
- lmfdb/lfunctions/templates/ellipticcurve.html (line 13)
- lmfdb/lfunctions/templates/genus2curve.html (line 12)

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

- 2020-05-04 09:42:44 by David Farmer (Reviewed)
- 2020-05-04 09:37:21 by David Farmer (Reviewed)
- 2019-05-03 05:34:03 by Andrew Sutherland (Reviewed)
- 2019-04-30 09:55:19 by Stephan Ehlen (Reviewed)
- 2019-04-30 08:15:57 by Stephan Ehlen (Reviewed)
- 2019-04-30 08:08:20 by Stephan Ehlen
- 2019-04-30 08:02:13 by Stephan Ehlen
- 2018-12-13 14:16:23 by Alex J. Best

**Differences**(show/hide)