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:

Holomorphic cusp form       Maass form       Elliptic curve       Artin representation


First complex
critical zero
Underlying
object
$N$ $\chi$ arithmetic self-dual $\nu$ $\delta_1,\delta_2$ $\mu$ $\varepsilon$
17.02494odd Maass1-1,19.53369-1
11.78454odd Maass2-1,15.417331
11.61497even Maass3-0,05.098741
9.22237holomorphic1-$\frac{11}{2}$1
7.21458K3 surface,
Hecke character,
holomorphic
7$\left(\frac{-7}{\cdot}\right)$$1$1
6.71631holomorphic5$\left(\frac{5}{\cdot}\right)$$\frac{5}{2}$$e(0.10108)$
6.56108holomorphic3$\left(\frac{-3}{\cdot}\right)$4$e(0.15625)$
6.50210even Maass10-0,04.58236-1
6.48044Calabi-Yau 3-fold,
holomorphic
6-$\frac{3}{2}$1
6.36261elliptic curve,
holomorphic
11-$\frac{1}{2}$1
5.10553odd Maass1-1,112.17300-1
4.06350holomorphic5$\left(\frac{5}{\cdot}\right)$$\frac{5}{2}$$e(0.89891)$
3.44334CM elliptic curve,
Hecke character,
holomorphic
36-$\frac{1}{2}$1
3.12943Maass5$\chi_5(2,\cdot)$0,14.86451$e(0.60295)$
3.02402Artin representation,
Hecke character,
even Maass
148$\left(\frac{37}{\cdot}\right)$0,001
2.89772even Maass1-0,013.779751
2.83254Artin representation,
odd Maass form
163?1,101
2.75562Artin representation,
Hecke character,
weight 1 cusp form
68$\left(\frac{-68}{\cdot}\right)$0,101
2.47446Artin representation,
Hecke character,
odd Maass form
136?1,101
1.73353rank 4 elliptic curve,
holomorphic
234446-$\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.