LMFDB - L-function 1-5520-5520.2069-r0-0-0

Dirichlet series

L(s) = 1

− 7^-s + i·11^-s + i·13^-s − 17^-s + i·19^-s + i·29^-s + 31^-s + i·37^-s + 41^-s + i·43^-s + 47^-s + 49^-s − i·53^-s − i·59^-s + i·61^-s + ⋯

L(s) = 1

− 7^-s + i·11^-s + i·13^-s − 17^-s + i·19^-s + i·29^-s + 31^-s + i·37^-s + 41^-s + i·43^-s + 47^-s + 49^-s − i·53^-s − i·59^-s + i·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$5520$ = $2^{4} \cdot 3 \cdot 5 \cdot 23$
Sign:	$-0.923 + 0.382i$
Analytic conductor:	$25.6347$
Root analytic conductor:	$25.6347$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5520} (2069, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 5520,\ (0:\ ),\ -0.923 + 0.382i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1823365676 + 0.9166678275i$
$L(\frac12)$	$\approx$	$0.1823365676 + 0.9166678275i$
$L(1)$	$\approx$	$0.8302591914 + 0.2566598649i$
$L(1)$	$\approx$	$0.8302591914 + 0.2566598649i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 $
	23	$ 1 $
good	7	$ 1 $
	11	$ 1 $
	13	$ 1 $
	17	$ 1 $
	19	$ 1 - T $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 + iT $
	41	$ 1 $
	43	$ 1 + iT $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 - T $
	67	$ 1 $
	71	$ 1 + iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 $
	97	$ 1 $
show more
show less

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

Imaginary part of the first few zeros on the critical line

−17.35528946793976196406792152060, −17.14179126931188459602513297460, −16.00966503783867387935517856025, −15.70974535877044890881876678312, −15.19627197910363274412892354013, −14.10192626616870743835467290118, −13.51152915948301855655294852457, −13.06824974916392663920300042593, −12.35888133649834092420768259340, −11.55348463829341462567323843150, −10.804729498720496618342218216232, −10.35569869713990322461548223255, −9.39276049944476900047507079590, −8.93638238954179186622362686358, −8.17557949300762658192829706453, −7.36787354331312158228083204671, −6.62099248418840221783891865765, −5.96752701037680346810198266673, −5.41104814882735304217539929966, −4.33722602261833112283284756914, −3.719182223388624192096968835894, −2.73229558338175487830161241011, −2.47774948428377192876693299920, −0.919142961321938077940390073148, −0.29553581535151486239942900130, 1.16509944842897096787992910128, 2.05349528853175634330821454303, 2.746471179648924834392746283776, 3.70313156829834407774369027568, 4.331420178001757408343009676747, 5.02334220627393431445764689383, 6.07629665187881760889291673061, 6.63788094557039594301959595577, 7.12248241477511602833127593104, 8.05451697301686470753499391195, 8.842609765434535128149715365891, 9.570122976675874008109020522729, 9.94465430308331038315446839587, 10.7981652112010482704402249259, 11.550610132375295147666115729776, 12.36602134218174568582405278927, 12.70321652323602268794770469598, 13.555584137890684057535382164314, 14.14636405551623279964975214915, 14.91363588876445130945125908796, 15.591094837408842136675932106168, 16.17441342283410299344930272223, 16.7983980475340703557492849556, 17.43866854959395101427898834906, 18.25060125060815816751278615670

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1823365676 + 0.9166678275i\)
\(L(\frac12)\)	\(\approx\)	\(0.1823365676 + 0.9166678275i\)
\(L(1)\)	\(\approx\)	\(0.8302591914 + 0.2566598649i\)
\(L(1)\)	\(\approx\)	\(0.8302591914 + 0.2566598649i\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more