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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 $\nu$ $\delta_1,\delta_2$ $\mu$ $\varepsilon$
17.02494odd Maass form111,19.53369-1
11.78454odd Maass form211,15.417331
11.61497even Maass form310,05.098741
9.22237holomorphic cusp form11$\frac{11}{2}$1
7.21458K3 surface,
Hecke character,
holomorphic cusp form
6.71631holomorphic cusp form5$\left(\frac{5}{\cdot}\right)$$\frac{5}{2}$$e(0.10108)$
6.56108holomorphic cusp form3$\left(\frac{-3}{\cdot}\right)$4$e(0.15625)$
6.50210even Maass form1010,04.58236-1
6.48044Calabi-Yau 3-fold,
holomorphic cusp form
6.36261elliptic curve,
holomorphic cusp form
5.10553odd Maass form111,112.17300-1
4.06350holomorphic cusp form5$\left(\frac{5}{\cdot}\right)$$\frac{5}{2}$$e(0.89891)$
3.44334CM elliptic curve,
Hecke character,
holomorphic cusp form
3.12943Maass form5$\chi_5(2,\cdot)$0,14.86451$e(0.60295)$
3.02402 Artin representation,
Hecke character,
even Maass form
148 $\left(\frac{37}{\cdot}\right)$0,001
2.89772even Maass form110,013.779751
2.83254Artin representation,
odd Maass form
2.75562Artin representation,
Hecke character,
weight 1 cusp form
2.47446Artin representation,
Hecke character,
odd Maass form
1.73353rank 4 elliptic curve,
holomorphic cusp form
0.12639Artin representation,
weight 1 cusp form
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 $\frac14$.