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Properties

Label 1-3024-3024.1019-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.00934 - 0.999i$
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 + (0.642 − 0.766i)13-s − 17-s + i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.642 + 0.766i)29-s + (0.939 − 0.342i)31-s + (0.866 − 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s + (−0.984 + 0.173i)11-s + (0.642 − 0.766i)13-s − 17-s + i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.642 + 0.766i)29-s + (0.939 − 0.342i)31-s + (0.866 − 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.00934 - 0.999i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.00934 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5260474613 - 0.5211534104i\)
\(L(\frac12)\) \(\approx\) \(0.5260474613 - 0.5211534104i\)
\(L(1)\) \(\approx\) \(0.7694770126 - 0.06932174524i\)
\(L(1)\) \(\approx\) \(0.7694770126 - 0.06932174524i\)

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 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−19.15591431993994519990720602550, −18.65000118358142040700043216514, −17.86817793097018251520651836769, −17.22126922817193980714801459101, −16.13778678117876819577332542642, −15.64316945003765996818583964342, −15.41435469659408903475133700101, −14.21546368033414815850684596146, −13.63411834525111220828931882183, −12.923323194241633680623561333174, −12.083805829286930260804927395638, −11.2100901543170642853188751283, −11.076227607922065142484609691027, −9.98887146064680086292000165641, −9.17620649496739047833160845233, −8.23299176590530627557693665486, −7.95440726680042641993681779023, −6.81607320420757147355631157596, −6.429759969909903860264721860880, −5.22168102149074392184974777277, −4.4888765227505031375846436015, −3.82965018928516658481474762309, −2.87828642555568581292003973248, −2.1471757167950533191727449065, −0.82415586341598238757615385219, 0.30005637657573140148960698766, 1.42603689141597268675356974772, 2.623119832798206510559965808312, 3.27643228154961462189789853301, 4.29909959619282261857515276616, 4.75110051200409950398828098736, 5.8478366823118888445183605656, 6.497471018920463504031350784999, 7.59615836985299271650624211486, 8.07662503588959866826009384166, 8.58779581572824018996022452604, 9.65965498571629454588263769650, 10.526234255168649374329483144927, 10.984942591189437563813033191223, 11.8441575237061902576452465982, 12.6067953062971181449684608844, 13.06734689653009220800037633926, 13.949612637985067051233883268811, 14.8945098732340163793926681090, 15.38331357128679246300077439180, 16.205638257129757946931910446910, 16.407168669857652002962724430835, 17.742296975055106206046959142414, 18.14165708999182111822817633967, 18.81531344841295256990839649810

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