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 level of the L-function, and $\chi$, known as the central character, is a primitive Dirichlet character of conductor dividing $N$. 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 search more complete list of degree 2 L-functions, or browse by underlying object:

 First complexcritical zero Underlyingobject $N$ $\chi$ arithmetic self-dual $\nu$ $\delta_1,\delta_2$ $\mu$ $\varepsilon$ 17.02494 odd Maass 1 - ○ ● 1,1 9.53369 -1 11.78454 odd Maass 2 - ○ ● 1,1 5.41733 1 11.61497 even Maass 3 - ○ ● 0,0 5.09874 1 9.22237 holomorphic 1 - ● ● $\frac{11}{2}$ 1 7.21458 K3 surface,Hecke character,holomorphic 7 $\left(\frac{-7}{\cdot}\right)$ ● ● $1$ 1 6.71631 holomorphic 5 $\left(\frac{5}{\cdot}\right)$ ● ○ $\frac{5}{2}$ $e(0.10108)$ 6.56108 holomorphic 3 $\left(\frac{-3}{\cdot}\right)$ ● ○ 4 $e(0.15625)$ 6.50210 even Maass 10 - ○ ● 0,0 4.58236 -1 6.48044 Calabi-Yau 3-fold,holomorphic 6 - ● ● $\frac{3}{2}$ 1 6.36261 elliptic curve,holomorphic 11 - ● ● $\frac{1}{2}$ 1 5.10553 odd Maass 1 - ○ ● 1,1 12.17300 -1 4.06350 holomorphic 5 $\left(\frac{5}{\cdot}\right)$ ● ○ $\frac{5}{2}$ $e(0.89891)$ 3.44334 CM elliptic curve,Hecke character,holomorphic 36 - ● ● $\frac{1}{2}$ 1 3.12943 Maass 5 $\chi_5(2,\cdot)$ ○ ○ 0,1 4.86451 $e(0.60295)$ 3.02402 Artin representation,Hecke character,even Maass 148 $\left(\frac{37}{\cdot}\right)$ ● ● 0,0 0 1 2.89772 even Maass 1 - ○ ● 0,0 13.77975 1 2.83254 Artin representation,odd Maass form 163 ? ● ○ 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 ? ● ● 1,1 0 1 1.73353 rank 4 elliptic curve,holomorphic 234446 - ● ● $\frac{1}{2}$ 1
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 $\frac14$, in which case the Maass form is holomorphic.