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Properties

Label 1-2960-2960.267-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.986 - 0.164i$
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  + (0.5 + 0.866i)3-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.866 + 0.5i)21-s + 23-s − 27-s + 29-s − i·31-s + (−0.866 + 0.5i)33-s + (−0.866 − 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.866 + 0.5i)21-s + 23-s − 27-s + 29-s − i·31-s + (−0.866 + 0.5i)33-s + (−0.866 − 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.629926511 - 0.2180608565i\)
\(L(\frac12)\) \(\approx\) \(2.629926511 - 0.2180608565i\)
\(L(1)\) \(\approx\) \(1.294124195 + 0.2752206937i\)
\(L(1)\) \(\approx\) \(1.294124195 + 0.2752206937i\)

Euler product

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

−19.00529368665977764237875991435, −18.39980900024413979322138375056, −17.48701355094291497878891304560, −17.14532888862435968355994386157, −16.17700173886804324328776125874, −15.09444365863288665359552892708, −14.644761156691849003622942284453, −14.205887814250679278546946448216, −13.13813711187432297192570789536, −12.728142418276240396366213040993, −11.88067561929545130758561210479, −11.33216307302652690017280097576, −10.417564945824254880333691829614, −9.52881541999979492177866089246, −8.5777436697617939091733732125, −8.14228136835141089807745271582, −7.64118245995859838191854804985, −6.5475438919836820852145067474, −5.9016432316109360839363617842, −5.168822081140196196212563907281, −4.10862795719913284912862104692, −3.08984366803931595524177968692, −2.509008153441017694495254826463, −1.51924658082200807349029203902, −0.84965161543874405317482275903, 0.43207019091910714255410645829, 1.721648624531597045388683936561, 2.46774469205680601358092044829, 3.31804250748447558748624965930, 4.39104178631850008346482361099, 4.80872174901987163488933821680, 5.31860228597653948397093112353, 6.94450355094574618868564050146, 7.22015839021962289397519434996, 8.208603304196150134960985440290, 8.95130223603375806009871235774, 9.58390307903613365749961556798, 10.3756635750388266586973143786, 10.86599425106798925177510904668, 11.8121498558008834508106145495, 12.39364106064605745511182531100, 13.660114207422361873573451128587, 13.95404430269017928237510261415, 14.95945270624089639791924427286, 15.07413584277298379537712239563, 16.08706217296059102630797147458, 16.845624595681800377759524840334, 17.393189230458964904932370575909, 18.03786511852010018051068171506, 19.1734537049791397435770629455

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