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Properties

Label 1-3024-3024.83-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.328 + 0.944i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8403300120 + 1.181731050i\)
\(L(\frac12)\) \(\approx\) \(0.8403300120 + 1.181731050i\)
\(L(1)\) \(\approx\) \(0.9414889815 + 0.1266492220i\)
\(L(1)\) \(\approx\) \(0.9414889815 + 0.1266492220i\)

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

−18.64964268605054758728649728375, −18.00522205620178112391150304456, −17.368757100745486880693254846012, −16.3430399554986923152551655268, −15.950327775422069820278782639667, −15.14303092469648855967096869953, −14.62595321695908376725361128397, −13.72213114215833759928414007725, −13.07860742615923365518381761486, −12.07121405260039030182925340086, −11.61810229795373960918906445638, −11.11305767910221509333085890812, −9.97311972649960544955596223084, −9.51386651888647621916473158587, −8.447999284100308304226715150994, −7.866262261418931566612028863597, −7.215419023944717031602354531, −6.36269999045644525142750645218, −5.56795708463927784052633688124, −4.580140854110571299227879489512, −3.85186175004967807530375354910, −3.23817909430257336358593186547, −2.24235303917118620790823412764, −1.043146427731500847608017679820, −0.30226326332126345654278548230, 0.9060983463044229114904520289, 1.63471028523873103818718880044, 2.85169498290702825379032830895, 3.7485946777429569981579267586, 4.27931220557987220872520843717, 5.02507228832336148570730965713, 6.27914276624667249577187902714, 6.68801427937104808522861525734, 7.604928223563458201461982736257, 8.411206284649923354747899471096, 8.95862929338986071009129328032, 9.70203183278721408540781297127, 10.810277980737742485181094726969, 11.33505117460625517283071083434, 11.99622874592235648039944980499, 12.57727470521153305196144046776, 13.5283819490937938328202481044, 14.25815840039704395143675904554, 14.87556849296288129133911334436, 15.673108060554272587803314746997, 16.34891569768150249669657063856, 16.77696785492805541161032258455, 17.75484729089362775824348146239, 18.43009025221486137171748191565, 19.261361954567753653069627349989

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