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L-functions can be organized by degree. All known degree 4 L-functions have a functional equation of one of the three forms: \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb C}(s + \nu_1) \, \Gamma_{\mathbb C}(s + \nu_2) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s+\mu_1) \, \Gamma_{\mathbb R}(s+\mu_2) \, \Gamma_{\mathbb C}\left(s+ \nu\right) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] or \[ \Lambda(s) := N^{s/2} \, \Gamma_{\mathbb R}(s+\mu_1) \, \Gamma_{\mathbb R}(s+\mu_2) \, \Gamma_{\mathbb R}(s+\mu_3) \, \Gamma_{\mathbb R}(s+\mu_4) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] and an Euler product of the form \[ L(s)= \prod_{p|N} \left(1-a_p\, p^{-s} + (a_p^2 - a_{p^2})\, p^{-2s} - (a_p^3 - 2 \, a_{p^2} \, a_p + a_{p^3} ) \, p^{-3s}\right)^{-1}. \prod_{p\nmid N} \left(1-a_p\, p^{-s} + (a_p^2 - a_{p^2})\, p^{-2s} - \chi(p) \, \overline{a_p} \, p^{-3s} +\chi(p) \, p^{-4s}\right)^{-1}. \] Here $N$ is the conductor of the L-function, and $\chi$, known as the central character, is a primitive Dirichlet character of conductor dividing $N$. Moreover, $\operatorname{Re}(\mu_j) \in \{0,1\}$, $\operatorname{Re}(\nu_j)$ and $\operatorname{Re}(\nu)$ are integer or an half-integer, and $2 \sum_j \mu_j + \sum_k \nu_k$ is a positive real.

Browse Degree 4 L-functions

A sample of degree 4 L-functions is in the table below. You can also browse by signature or underlying object:

Signature (0,0,0,0;)     Symmetric cube of L-function of elliptic curve     Artin representation of dimension 4     Genus 2 curve over $\Q$     Hilbert modular form over a quadratic field     Elliptic curve over a quadratic field

First complex
critical zero
underlying object $N$ $\chi$ arithmetic self-dual $\Gamma_{\R}$ parameters $\Gamma_{\C}$ parameters $\varepsilon$
16.18901$\SL(4,\Z)$ Maass form11$16.89i$, $2.272i$, $-6.035i$, $-13.13i$1
14.49606$\Sp(4,\Z)$ Maass form11$4.720i$, $-4.720i$, $12.46i$, $-12.46i$1
4.30352genus 2 curve paramodular form2771$\frac12$, $\frac12$1
3.67899Hilbert cusp form over $\Q(\sqrt{5})$Elliptic curve over $\Q(\sqrt5)$7751$\frac12$, $\frac12$1
3.50464Artin representation1609$\left(\frac{1609}{\cdot}\right)$0, 0, 1, 11
2.78027Artin representation3215$\left(\frac{-3215}{\cdot}\right)$0, 1, 1, 11
2.48044Sp(4,$\mathbb Z$) Maass form11$19.81i$, $-19.81i$, $8.531i$, $-8.531i$1
2.26074CM genus 2 curveCM Hilbert cusp form over $\Q(\sqrt{5})$31251$\frac12$, $\frac12$1
2.32002$\Sym^3$ of an elliptic curve 32151$\frac32$, $\frac12$1
1.42931RM genus 2 curveHilbert cusp form over $\Q(\sqrt{5})$125001$\frac12$, $\frac12$1
1.21844QM genus 2 curveBianchi cusp form over $\Q(\sqrt{-3})$207361$\frac12$, $\frac12$1