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Properties

Label 1-3024-3024.2461-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.925 + 0.377i$
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  + (−0.642 − 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (0.766 − 0.642i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.766 + 0.642i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (0.766 − 0.642i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.766 + 0.642i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482622293 + 0.2909085134i\)
\(L(\frac12)\) \(\approx\) \(1.482622293 + 0.2909085134i\)
\(L(1)\) \(\approx\) \(1.039007487 - 0.03536848359i\)
\(L(1)\) \(\approx\) \(1.039007487 - 0.03536848359i\)

Euler product

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

−18.84533352284869232696946784436, −18.46554689569631869971042662777, −17.63056919467131504926063107560, −17.04924272622221549203059772984, −15.983507519846180013877019208741, −15.49827639732259296544611724472, −14.93238211410810311380388871223, −14.11583726464915260216664922757, −13.50482237189971077604994894491, −12.568507198898172350675776450392, −11.785432766308781900012179832577, −11.37122828794435528740097116331, −10.43133449190301877246484146641, −9.90418312308684754562848729015, −8.93216563906757148912917745138, −8.21455529722616348913326435915, −7.28413440641133650239243831639, −6.92340453394158438859895432746, −6.01340124136881083228778764609, −5.062778006752833833475881087640, −4.24033571817943606965258692569, −3.40671908663433144005659171301, −2.80210970818206513721848431037, −1.71541169860991696520368102156, −0.578487209966920790330741086079, 0.95873574777014378185377938130, 1.51857140125348615680331548458, 2.86346231011377545081735469337, 3.75275898708389164371666844130, 4.2811497666497857807803917450, 5.19059764376426042373414540107, 6.011695432746957596089748502352, 6.80340934464139022955422318778, 7.63813382948262487416415719347, 8.46551098482283782147145054614, 9.03613884085159490958284434914, 9.54065583892113052507922998332, 10.928477140465122566335715092944, 11.26050683373433104996563238138, 11.94091547507485078363229723292, 12.83440707551357166384932016520, 13.38150821705306909830840560753, 14.14138744268022762110668314522, 14.97760056960994283003900775453, 15.73356886492978400660284885931, 16.284301746887542899553562857479, 16.93923176906041554773358577358, 17.53527451648220440032912062706, 18.574239978734356130517956069217, 19.17003588753817147756726400425

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