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Properties

Label 1-287-287.156-r0-0-0
Degree 11
Conductor 287287
Sign 0.758+0.651i0.758 + 0.651i
Analytic cond. 1.332821.33282
Root an. cond. 1.332821.33282
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(287s/2 ΓR(s) L(s)=((0.758+0.651i) Λ‾(1−s)\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}
Λ(s)=(287s/2 ΓR(s) L(s)=((0.758+0.651i) Λ‾(1−s)\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: 11
Conductor: 287287    =    7⋅417 \cdot 41
Sign: 0.758+0.651i0.758 + 0.651i
Analytic conductor: 1.332821.33282
Root analytic conductor: 1.332821.33282
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ287(156,⋅)\chi_{287} (156, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 287, (0: ), 0.758+0.651i)(1,\ 287,\ (0:\ ),\ 0.758 + 0.651i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9826220703+0.3642379090i0.9826220703 + 0.3642379090i
L(12)L(\frac12) ≈\approx 0.9826220703+0.3642379090i0.9826220703 + 0.3642379090i
L(1)L(1) ≈\approx 0.8832828299+0.2893310142i0.8832828299 + 0.2893310142i
L(1)L(1) ≈\approx 0.8832828299+0.2893310142i0.8832828299 + 0.2893310142i

Euler product

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