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Properties

Label 1-3024-3024.1795-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.837 - 0.546i$
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  + (0.342 − 0.939i)5-s + (−0.342 − 0.939i)11-s + (−0.342 + 0.939i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.342 − 0.939i)29-s + (−0.939 − 0.342i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.939 + 0.342i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (−0.342 − 0.939i)11-s + (−0.342 + 0.939i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.342 − 0.939i)29-s + (−0.939 − 0.342i)31-s − i·37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.939 + 0.342i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.837 - 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3227912917 - 1.086396198i\)
\(L(\frac12)\) \(\approx\) \(0.3227912917 - 1.086396198i\)
\(L(1)\) \(\approx\) \(0.9433859247 - 0.3406064786i\)
\(L(1)\) \(\approx\) \(0.9433859247 - 0.3406064786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.342 - 0.939i)T \)
11 \( 1 + (-0.342 - 0.939i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
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 \( 1 + T \)
79 \( 1 + (-0.173 + 0.984i)T \)
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

−19.24885371974883403718311101064, −18.42557870862395882054924914068, −18.11124754760130452025632618993, −17.32583123886247233832927836102, −16.65494211143358118696929838551, −15.70243671716527328681073962602, −14.97205635839179438647583813930, −14.589808525389557663747922066848, −13.86102219271562225341804548180, −12.749377086430252540176462772260, −12.5690536198701606416453882386, −11.46305404949084397831672544537, −10.66893505397849764797286703780, −10.111999924006478748495868001249, −9.6467804328430670967621779852, −8.481350097433503263200009386425, −7.72237013695447572164116885756, −7.1074502092500392373674761253, −6.34523812631531632011815242589, −5.47594437923353359811793716291, −4.8704336221857992019819002530, −3.62968013126059930025130879727, −3.08412667997010305340190577259, −2.17073662282415437681882035612, −1.34597180915623727405599486027, 0.33869265695367273019055928428, 1.32922715354296031036156940631, 2.22380605621566783487374078303, 3.20316071096298107514345668411, 4.06689643407651343633998957969, 5.04149032743157123716379891605, 5.479727687403370528823389691256, 6.30381215544310362485855778075, 7.402561784690543987682370536081, 7.87922582359892147068268580179, 9.02157164381198742821524141897, 9.30165366264842383125663240659, 10.01894114998436302020515203267, 11.253362106419974770336718876211, 11.55854580351168266555147576008, 12.46563895237032118110798651831, 13.20603056509972317684998033938, 13.86873375344924619139427369128, 14.2764742341729953925383656915, 15.525458341777252414237310388439, 16.042725802494050215762753624581, 16.657579807903912341622081960351, 17.250816512458597088501554939989, 18.06340046365840031000080347138, 18.79546115888497226881678779784

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