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Properties

Label 1-3024-3024.2083-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.377 - 0.925i$
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  + (−0.642 − 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

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1660427202 - 0.1115714466i\)
\(L(\frac12)\) \(\approx\) \(0.1660427202 - 0.1115714466i\)
\(L(1)\) \(\approx\) \(0.6807806406 + 0.05951055928i\)
\(L(1)\) \(\approx\) \(0.6807806406 + 0.05951055928i\)

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

−18.89006917951364412492407473539, −18.382532789379209991077372630631, −17.81685093002936139740151457481, −16.87899104337235711092472505533, −16.007491432187372266597543626465, −15.53315731625143507888037931987, −14.986369456370424858047745110975, −14.04039246752842361150101512491, −13.41142284180452619026504956937, −12.79127056705781029521695253348, −11.69033042282847231996403361287, −11.21000443877457640906370388785, −10.65345820296892209265921276317, −9.85050300770494852231072319715, −8.91747089206382597037957064601, −7.97094951530718554666716353878, −7.7510342639253330755200359010, −6.58269825887775891716411714051, −6.09728132821123095409762663495, −5.1200025998115076407772799846, −4.2216096320659826388143873816, −3.35622682752181555125037502063, −2.80891000060725547682845764477, −1.81904561809256587469823857879, −0.38333748430178030822241752324, 0.06810843675400373472108737950, 1.59169537961277702917477388427, 1.941166413611357144620166298957, 3.378701744125009247450509400741, 4.10424540450629777721845658016, 4.66145097295039479680325068629, 5.57948115699248147712427739174, 6.41661842101805025684745991858, 7.276888376495887011241121225235, 8.03591468797578127627567516494, 8.702196049254857696320920391453, 9.25759257457510768005548951139, 10.383375286599384490895554844076, 10.83717775250300424346841115400, 11.8235715593401665886035546218, 12.4363415471681085506619669571, 12.98007824280851136111360914232, 13.70625947806901203401953165854, 14.78942228630438715845964406371, 15.15839152888897149083252535555, 16.21795811414760785939455056401, 16.38119315543172747273347094052, 17.323895372167621664186269295484, 18.08503465361796181095473304750, 18.75429321412792515917645997060

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