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Properties

Label 1-3024-3024.1483-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.900 - 0.435i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 + 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (0.766 + 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.766 − 0.642i)47-s + i·53-s − 55-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 + 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (0.766 + 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.766 − 0.642i)47-s + i·53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.900 - 0.435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.318078073 - 0.3022837302i\)
\(L(\frac12)\) \(\approx\) \(1.318078073 - 0.3022837302i\)
\(L(1)\) \(\approx\) \(0.9513327573 - 0.08233120125i\)
\(L(1)\) \(\approx\) \(0.9513327573 - 0.08233120125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (0.984 - 0.173i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.12981661833877468506208060831, −18.819660106634219791525365441014, −17.38248052972057831672002261852, −17.13862039819483612394316074454, −16.34633000880424972870231781039, −15.524661160836006637353259765054, −14.820569651363612828999073025849, −14.47618657784937880421499778679, −13.469507550171569301403043507135, −12.61270257221098249793542144802, −11.9588170933493316799485505030, −11.44324449880788457584030821355, −10.67647644460838204792349552495, −9.85574041926454366187261165660, −8.94486408843959678325803415514, −8.41681598790132908845957223944, −7.546202094917078077728606326292, −6.74219180884134360573221006943, −6.32202561570429431387008162013, −5.06022667570101083303102036875, −4.183052036476865637851621017726, −3.87651654204296224089754173937, −2.74798985970080983054773630454, −1.84657177193531936335232186832, −0.72980542695672790888309687648, 0.642277461171376336894371548881, 1.504092522224184406711974743196, 2.84909238114231472592845404405, 3.466781539119555123425228974185, 4.23936497540419023256516025520, 5.09998126246233432763890193000, 5.80317457877550338195643457348, 7.02671115488570550510430460520, 7.28080041106570671152296637843, 8.364452977259115073980741060815, 8.800300767827928847046624015456, 9.715328667386820800942421543133, 10.59958983660086994309975202497, 11.21350379089369930067960109693, 12.20024473729115847666155674913, 12.29894026108710659883395575772, 13.36284375794293278355415066735, 14.20080107064992445665256461770, 14.90519190551968119740385179550, 15.43361044601436781893119928003, 16.25074584063576714247042678418, 16.85712758313786151328192933308, 17.511215650509949853007217100723, 18.39316849108660148338885489989, 19.227438733295525525133626551007

Graph of the $Z$-function along the critical line