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Properties

Label 1-287-287.156-r0-0-0
Degree $1$
Conductor $287$
Sign $0.758 + 0.651i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.669 + 0.743i)2-s + (0.866 + 0.5i)3-s + (−0.104 − 0.994i)4-s + (−0.913 − 0.406i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 + 0.866i)9-s + (0.913 − 0.406i)10-s + (−0.406 − 0.913i)11-s + (0.406 − 0.913i)12-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)15-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.978 − 0.207i)18-s + (−0.207 − 0.978i)19-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.866 + 0.5i)3-s + (−0.104 − 0.994i)4-s + (−0.913 − 0.406i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 + 0.866i)9-s + (0.913 − 0.406i)10-s + (−0.406 − 0.913i)11-s + (0.406 − 0.913i)12-s + (0.951 − 0.309i)13-s + (−0.587 − 0.809i)15-s + (−0.978 + 0.207i)16-s + (0.406 + 0.913i)17-s + (−0.978 − 0.207i)18-s + (−0.207 − 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9826220703 + 0.3642379090i\)
\(L(\frac12)\) \(\approx\) \(0.9826220703 + 0.3642379090i\)
\(L(1)\) \(\approx\) \(0.8832828299 + 0.2893310142i\)
\(L(1)\) \(\approx\) \(0.8832828299 + 0.2893310142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.743 + 0.669i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.994 - 0.104i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.207 + 0.978i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−25.525838555480443116388361612774, −24.95275717084362688976615501490, −23.30050200519005570866588844595, −23.00311854193880246286643317635, −21.35991011946512507738743491723, −20.657176731442049178569373000732, −19.92572293294900936611062715090, −18.98764966469063496479286466373, −18.51968269829331915986426817682, −17.622302738678819251619433838194, −16.17418326770187771769726165031, −15.376702754657860475349401245752, −14.19582334218064871035501743383, −13.17226265193486193842071511258, −12.2181446256930079689513178511, −11.47413147563769706048294891297, −10.25003625319984728613601430875, −9.32387465160582201237496478413, −8.18968637956649063045817325992, −7.62682840589742680015319253465, −6.66075630950929936899245304041, −4.40361559961629218166792205481, −3.43422750309537889786190132489, −2.51176345642359403899099439991, −1.178032525949558605567819869660, 1.0467235850562963197039817448, 2.9096845297987150441949789607, 4.16174303024866779236104213893, 5.21495736959170880333819533894, 6.58641004872623845344873380389, 7.935226406767015717024039482515, 8.39669499774701759256210930362, 9.13775059527889761112411396061, 10.51562830871314434539477655107, 11.11472643127606437838808520734, 12.88888124452752990032455158882, 13.814239350665876160985005835393, 14.914103106777635137282999076802, 15.61760626257075228374106161396, 16.2248234194494489253646976898, 17.15042010632294633165633614546, 18.65284898733115781525173027652, 19.12427934766062347137713846949, 20.05458491924488094735036634214, 20.81655110187501126996821988416, 22.00325392084744075526254515655, 23.35077083484328502765906185071, 23.94466207150222561609510762268, 24.87864478506094788895257643309, 25.76449988146632786175870213602

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