The **conductor** of an L-function is the integer $N$ occurring in its functional equation

\[ \Lambda(s) := N^{s/2} \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). \]

The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$.

In the literature, the word *level* is sometimes used instead of *conductor*.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Christelle Vincent on 2015-09-16 18:40:07

**Referred to by:**

- intro.tutorial
- lfunction
- lfunction.analytic_conductor
- lfunction.bad_prime
- lfunction.central_character
- lfunction.degree
- lfunction.euler_product
- lfunction.functional_equation
- lfunction.invariants
- lfunction.label
- lfunction.selbergdata
- rcs.cande.lfunction
- rcs.rigor.lfunction.curve
- lmfdb/lfunctions/templates/Degree2.html (line 26)
- lmfdb/lfunctions/templates/Degree2.html (line 73)
- lmfdb/lfunctions/templates/Degree3.html (line 28)
- lmfdb/lfunctions/templates/Degree3.html (line 69)
- lmfdb/lfunctions/templates/Degree4.html (line 36)
- lmfdb/lfunctions/templates/Degree4.html (line 82)
- lmfdb/lfunctions/templates/Lfunction.html (line 82)

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

- 2015-09-16 18:40:07 by Christelle Vincent (Reviewed)