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Properties

Label 1-3024-3024.139-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.187 + 0.982i$
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 + (0.642 + 0.766i)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 + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (0.173 − 0.984i)47-s + i·53-s − 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.642 + 0.766i)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 + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (0.173 − 0.984i)47-s + i·53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191732478 + 1.441410599i\)
\(L(\frac12)\) \(\approx\) \(1.191732478 + 1.441410599i\)
\(L(1)\) \(\approx\) \(1.081796539 + 0.4405748951i\)
\(L(1)\) \(\approx\) \(1.081796539 + 0.4405748951i\)

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 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + iT \)
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 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
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

−18.83374678990718284107157781648, −18.20973320624890861289883485112, −17.41797844295649708205494838267, −16.57309916693304244516735413339, −16.14594579724434110804696431112, −15.59313557151110241258437232574, −14.59924454977140043465562168485, −13.87136288145303590694493215901, −13.17723513306259335829881065778, −12.59432579646573211586832222525, −11.660872394425866406877835662673, −11.29206038317288813550689300947, −10.3053668380681129954137927211, −9.43264899061097130828352456120, −8.79816460179097395405791409396, −8.1438634043969694550593804978, −7.48704636186200509560604356886, −6.44042272865477072300251745575, −5.645357231022168618049730823821, −5.00689290194831518975579137096, −4.16032107562961059041585542242, −3.29921949033787093860467688032, −2.57479169099365049993554890544, −1.02433294048796444869290015239, −0.77742762350851194276247447132, 1.18066318074335977605097609311, 2.00214585326148777311953713661, 2.993577647342910683303393512467, 3.81735760935161832684608534963, 4.33892423415830767016911401268, 5.55436545432949932915388084666, 6.253040354579585327930279428536, 7.036290668118162171297122405180, 7.60794981445896114434952690128, 8.40234363891077832148125748289, 9.41571321566179810753533980714, 9.993224569138557708672751763234, 10.74039946783809509745247242030, 11.5459458761467351486792990783, 12.01852301658298605828317960830, 12.87044289258768640404848035781, 13.83601997955958937003241612521, 14.37159457168661140648590590772, 14.9763710744783307904461766816, 15.77447942755322792380467401200, 16.282213383031153830223827825149, 17.39954018344337098412637736163, 17.78051949491509852332204256166, 18.62708006177473707475851008989, 19.21747464742235347012008854925

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