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Properties

Label 1-3024-3024.187-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: $\chi_{3024} (187, \cdot )$
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

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

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