LMFDB - L-function 4-1280e2-1.1-c0e2-0-6

Dirichlet series

L(s) = 1

+ 2·5^-s + 3·25^-s − 4·61^-s − 81^-s − 4·89^-s + 4·109^-s + 2·121^-s + 4·125^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 2·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + 211^-s + 223^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$1638400$ = $2^{16} \cdot 5^{2}$
Sign:	$1$
Analytic conductor:	$0.408069$
Root analytic conductor:	$0.799251$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 1638400,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.584771535$
$L(\frac12)$	$\approx$	$1.584771535$
$L(1)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2		$ 1 $
	5	$C_1$	$ ( 1 - T )^{2} $
good	3	$C_2^2$	$ 1 + T^{4} $
	7	$C_2^2$	$ 1 + T^{4} $
	11	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	13	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	17	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	19	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	23	$C_2^2$	$ 1 + T^{4} $
	29	$C_2$	$ ( 1 + T^{2} )^{2} $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	41	$C_2$	$ ( 1 + T^{2} )^{2} $
	43	$C_2^2$	$ 1 + T^{4} $
	47	$C_2^2$	$ 1 + T^{4} $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	61	$C_1$	$ ( 1 + T )^{4} $
	67	$C_2^2$	$ 1 + T^{4} $
	71	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	73	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	79	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	83	$C_2^2$	$ 1 + T^{4} $
	89	$C_1$	$ ( 1 + T )^{4} $
	97	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.11203810174608963954547733556, −9.755788897299839985655381272396, −9.169501381155814526192066296346, −9.056298990568805909254464998223, −8.567059092909128343430148032235, −8.196964209605698150821418862154, −7.39127284259984048355575565345, −7.30194741125382820024151431869, −6.66210640114049580428996427701, −6.25364858686086875291811738442, −5.84735993913501035945842573364, −5.69341348208068265450505147342, −5.00693055402881876529990933414, −4.65716182880845753263469192982, −4.19828035913590547031411633613, −3.23560474017522180276188755746, −2.97302131512843998398094059141, −2.33929085435462261873416935934, −1.72793240037226587624073818307, −1.26909857394861531297243792085, 1.26909857394861531297243792085, 1.72793240037226587624073818307, 2.33929085435462261873416935934, 2.97302131512843998398094059141, 3.23560474017522180276188755746, 4.19828035913590547031411633613, 4.65716182880845753263469192982, 5.00693055402881876529990933414, 5.69341348208068265450505147342, 5.84735993913501035945842573364, 6.25364858686086875291811738442, 6.66210640114049580428996427701, 7.30194741125382820024151431869, 7.39127284259984048355575565345, 8.196964209605698150821418862154, 8.567059092909128343430148032235, 9.056298990568805909254464998223, 9.169501381155814526192066296346, 9.755788897299839985655381272396, 10.11203810174608963954547733556

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.584771535\)
\(L(\frac12)\)	\(\approx\)	\(1.584771535\)
\(L(1)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more