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Properties

Label 1-2960-2960.2403-r1-0-0
Degree $1$
Conductor $2960$
Sign $-0.996 + 0.0817i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)7-s + (−0.939 + 0.342i)9-s + (0.866 + 0.5i)11-s + (−0.342 + 0.939i)13-s + (−0.939 + 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.642 + 0.766i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)27-s + (0.5 − 0.866i)29-s − i·31-s + (0.342 − 0.939i)33-s + (0.984 + 0.173i)39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)7-s + (−0.939 + 0.342i)9-s + (0.866 + 0.5i)11-s + (−0.342 + 0.939i)13-s + (−0.939 + 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.642 + 0.766i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)27-s + (0.5 − 0.866i)29-s − i·31-s + (0.342 − 0.939i)33-s + (0.984 + 0.173i)39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.996 + 0.0817i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ -0.996 + 0.0817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02319289359 - 0.5666663383i\)
\(L(\frac12)\) \(\approx\) \(0.02319289359 - 0.5666663383i\)
\(L(1)\) \(\approx\) \(0.7868053467 - 0.2709095873i\)
\(L(1)\) \(\approx\) \(0.7868053467 - 0.2709095873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.642 - 0.766i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.385586031623017116488383350202, −18.57101184524543692687147951393, −17.819389936438106047952471674996, −17.00711161334659628443275879638, −16.40712463640189554772763205290, −15.87287629364130910095105333890, −14.958663845513645370454581705103, −14.74439102807299409937037722817, −13.71512394302653977021171661267, −12.82659142475655192690366204225, −12.123340425931987361302312019765, −11.46064764589601315662418640204, −10.65129815362584061849392280063, −10.02229730843017876980940054144, −9.22819892607867790838095469706, −8.752283627749739003635690798965, −7.977393055797114845340497349243, −6.65066883891312794406376715869, −6.162630222153754044630966986249, −5.37071893934652255304005532455, −4.62573261391766845527187169245, −3.70729174306103915018447436958, −3.05097936410373511142812439374, −2.27612567981409916363996898668, −0.811573102592611826639398429431, 0.12135208010041391233994411641, 1.05844612535053817343710603996, 1.90113119401906843557937962640, 2.70732481842810500315361272918, 3.80545456386906173652722282869, 4.49410956629028599828033512715, 5.49388155901432987877157930113, 6.60513365554218249484598398817, 6.85082268863697399966609825772, 7.3697433053598794109290753466, 8.58950537261504438264835000452, 9.09932093263769356646089111706, 9.975541276140069559469547475038, 10.88698352315733838591477190031, 11.574847831380353042793994362097, 12.21588075767432810009697734000, 12.93225572644481944009434947724, 13.78555860218789751729061277730, 13.90379313275932622257357872363, 15.07192447895556658817956009331, 15.73228790674840410410411135692, 16.87167441024158232971823512946, 17.14204746522984879182305713170, 17.69657378358861358869380632586, 18.67756829663230171323938980363

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