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Properties

Label 1-3024-3024.2867-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  + (−0.984 + 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.984 + 0.173i)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 + (−0.984 − 0.173i)29-s + (−0.173 − 0.984i)31-s − i·37-s + (0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (0.173 − 0.984i)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 + (−0.984 + 0.173i)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 + (−0.984 − 0.173i)29-s + (−0.173 − 0.984i)31-s − i·37-s + (0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (0.173 − 0.984i)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: $\chi_{3024} (2867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6508898191 + 0.1277124932i\)
\(L(\frac12)\) \(\approx\) \(0.6508898191 + 0.1277124932i\)
\(L(1)\) \(\approx\) \(0.6902685355 + 0.01139153861i\)
\(L(1)\) \(\approx\) \(0.6902685355 + 0.01139153861i\)

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

−19.06628004562250018415702403111, −18.46059744275220266508951439715, −17.53469632745817196089730230918, −16.88612825369298587495600182557, −16.20169972664867651631591340212, −15.40420183486485249756481552053, −14.98592698247192178264151017705, −14.25253556689368994730674765322, −13.1998287026351214124648011501, −12.49624090219521390415316811012, −12.19361125487063104617894720017, −11.14240807678696203416382961633, −10.51313808300198056714715696594, −9.90633282162194778223429525664, −8.75495730655143127967534744187, −8.25442948526528340650788152483, −7.495541666396481797752469333, −6.92347331006803100401430221521, −5.829624293774231424795556128322, −5.01059419194773318857085691017, −4.363563458234371689709570449267, −3.51740287316129492727202543729, −2.65549551637738104731818026069, −1.772378185668987180398356356265, −0.38891935380254801138319523880, 0.500819968258193878084353000636, 1.98634649054404921893954258349, 2.730441643426796459320248364751, 3.63342171135889975765969128311, 4.35235295670206235783876747128, 5.20039428438577298343544454824, 5.93135020435879449898025869998, 7.04663331917511819533540226885, 7.649325688815556596757604502523, 8.074043384424184375227629632137, 9.08864665414240517790905847869, 9.93512150530805268617631427185, 10.53642021707869207585007542926, 11.536004303906475277559184655912, 11.82255075261911438902238053063, 12.790114527604672499901233631286, 13.32766585627344671186277548990, 14.4259463416576665137320119433, 14.873461942103507165959297436888, 15.581589463789244727207018167047, 16.38194015629120813995373085104, 16.74570300121327761203518464696, 17.85423969992247302473297851715, 18.50685174551321062323906126044, 19.0433398362085502400753321625

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