L-functions can be organized by degree.
All known degree 2 L-functions have a
functional equation of one of the two forms
\[
\Lambda(s) := N^{s/2}
\Gamma_{\mathbb C}(s+\nu)
\cdot L(s) = \varepsilon \overline{\Lambda}(1-s)
\]
or
\[
\Lambda(s) := N^{s/2}
\Gamma_{\mathbb R}(s+\delta_1+ i \mu)
\Gamma_{\mathbb R}(s+\delta_2- i \mu)
\cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
\]
and an Euler product of the form
\[
L(s)=
\prod_{p|N} \left(1-a_p p^{-s}\right)^{-1}.
\prod_{p\nmid N} \left(1-a_p p^{-s}+\chi(p) p^{-2s}\right)^{-1}.
\]
Here $N$ is the conductor of the L-function,
$\varepsilon$ is the sign,
and $\chi$, known as the central character,
is a primitive Dirichlet character of conductor dividing $N$.
The functions $\Gamma_\R$ and $\Gamma_\C$
are the gamma factors.
The parameter $\nu$ is a positive integer or half-integer, the parameter
$\mu$ is real, and
$\delta_j$ is 0 or 1.
If the central character is even (or odd) then either $2\nu+1$ or $\delta_1+\delta_2$ is
even (or odd).

## Browse Degree 2 L-functions

A sampling of degree 2 L-functions is in the table below. You can also
browse
by underlying object:

Holomorphic cusp form Maass form Elliptic curve Artin representation

First complex critical zero |
underlying object | $N$ | $\chi$ | arithmetic | self-dual | $\nu$ | $\delta_1,\delta_2$ | $\mu$ | $\varepsilon$ |
---|---|---|---|---|---|---|---|---|---|

17.02494 | odd Maass form | 1 | 1 | ✔ | 1,1 | 9.53369 | -1 | ||

11.78454 | odd Maass form | 2 | 1 | ✔ | 1,1 | 5.41733 | 1 | ||

11.61497 | even Maass form | 3 | 1 | ✔ | 0,0 | 5.09874 | 1 | ||

9.22237 | holomorphic cusp form | 1 | 1 | ✔ | ✔ | $\frac{11}{2}$ | 1 | ||

7.21458 | K3 surface, Hecke character, holomorphic cusp form | 7 | $\left(\frac{-7}{\cdot}\right)$ | ✔ | ✔ | $1$ | 1 | ||

6.71631 | holomorphic cusp form | 5 | $\left(\frac{5}{\cdot}\right)$ | ✔ | $\frac{5}{2}$ | $e(0.10108)$ | |||

6.56108 | holomorphic cusp form | 3 | $\left(\frac{-3}{\cdot}\right)$ | ✔ | 4 | $e(0.15625)$ | |||

6.50210 | even Maass form | 10 | 1 | ✔ | 0,0 | 4.58236 | -1 | ||

6.48044 | Calabi-Yau 3-fold, holomorphic cusp form | 6 | 1 | ✔ | ✔ | $\frac{3}{2}$ | 1 | ||

6.36261 | elliptic curve, holomorphic cusp form | 11 | 1 | ✔ | ✔ | $\frac{1}{2}$ | 1 | ||

5.10553 | odd Maass form | 1 | 1 | ✔ | 1,1 | 12.17300 | -1 | ||

4.06350 | holomorphic cusp form | 5 | $\left(\frac{5}{\cdot}\right)$ | ✔ | $\frac{5}{2}$ | $e(0.89891)$ | |||

3.44334 | CM elliptic curve, Hecke character, holomorphic cusp form | 36 | 1 | ✔ | ✔ | $\frac{1}{2}$ | 1 | ||

3.12943 | Maass form | 5 | $\chi_5(2,\cdot)$ | 0,1 | 4.86451 | $e(0.60295)$ | |||

3.02402 | Artin representation, Hecke character, even Maass form | 148 | $\left(\frac{37}{\cdot}\right)$ | ✔ | ✔ | 0,0 | 0 | 1 | |

2.89772 | even Maass form | 1 | 1 | ✔ | 0,0 | 13.77975 | 1 | ||

2.83254 | Artin representation, odd Maass form | 163 | $\chi_{163}(104,\cdot)$ | ✔ | 1,1 | 0 | 1 | ||

2.75562 | Artin representation, Hecke character, weight 1 cusp form | 68 | $\left(\frac{-68}{\cdot}\right)$ | ✔ | ✔ | 0,1 | 0 | 1 | |

2.47446 | Artin representation, Hecke character, odd Maass form | 136 | $\left(\frac{136}{\cdot}\right)$ | ✔ | ✔ | 1,1 | 0 | 1 | |

1.73353 | rank 4 elliptic curve, holomorphic cusp form | 234446 | 1 | ✔ | ✔ | $\frac{1}{2}$ | 1 | ||

0.12639 | Artin representation, weight 1 cusp form | 3997 | $\chi_{3997}(13,\cdot)$ | ✔ | 0,1 | 0 | $e(-0.0742)$ |

The above table is intended to illustrate the variety of properties which degree 2 L-functions
can have. Several constraints are illustrated, such as:

- The $\Gamma$-factors determine the parity of the central character.
- A self-dual degree 2 L-function with nontrivial character must be CM, in particular at least half its prime coefficients $a_p$ must vanish.
- L-functions of Maass forms are (conjecturally) non-arithmetic unless the Maass form has eigenvalue $\frac14$.