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Properties

Label 1-3024-3024.283-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.978 + 0.206i$
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 + (0.342 − 0.939i)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 + (0.342 + 0.939i)29-s + (−0.939 − 0.342i)31-s + i·37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.939 + 0.342i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.342 − 0.939i)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 + (0.342 + 0.939i)29-s + (−0.939 − 0.342i)31-s + i·37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.939 + 0.342i)47-s + (0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.978 + 0.206i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.978 + 0.206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06942129746 + 0.6658423826i\)
\(L(\frac12)\) \(\approx\) \(0.06942129746 + 0.6658423826i\)
\(L(1)\) \(\approx\) \(0.8386127042 + 0.2457854988i\)
\(L(1)\) \(\approx\) \(0.8386127042 + 0.2457854988i\)

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

−18.96534155832470961980912968434, −18.0289812272138718617022008992, −17.09044543354231712694907667347, −16.54826220612627265893153435238, −16.23884779474691001495179289977, −15.1804720844064485906244558042, −14.59976998771294530477144332029, −13.681224036958806941773756043775, −13.10947384619641474175929765229, −12.37249402994207833567911427877, −11.66911805836939270753878209011, −11.042463698126970108170470401290, −10.20375200532577165185280580298, −9.23743763344731752667897583192, −8.55308981460522104500053880344, −8.29211537376367915646721316126, −7.121482912268733518485126644118, −6.32065914771680056492683057114, −5.65099685756392580599961064002, −4.660873103843601366544123056437, −4.05754720773265086370894957135, −3.32247497209536450974625059495, −2.07493185543283636656113052811, −1.29371268397845093336008338664, −0.207070536111287407470829239937, 1.32012599370718225763565712975, 2.23363099438199262201088246599, 3.24695270894605767949750592713, 3.67278378558045146020170540854, 4.78714267659684678653770467107, 5.53157576118277995905206101999, 6.511202596961796570460189966445, 7.09918625633101089475532906361, 7.78074472817671323431935030416, 8.52657219952016013212116989639, 9.58545507197326483183534359186, 10.16510828216805467177888760252, 10.8155599822722608381179694951, 11.639179979967987576599438344186, 12.20896610998090476255420818972, 13.07219555337979249664887396068, 13.79839553558711241080017621868, 14.73828179333997060352442402673, 15.01572869946255281696321091772, 15.74330047168978388299469519652, 16.61222376674575102721007327751, 17.35552198940607871600044812505, 18.26570712047270489193768471442, 18.381824653254939057074633467627, 19.47075395496784226257171289002

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