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Properties

Label 1-1287-1287.119-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.775 - 0.631i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.994 − 0.104i)5-s + (0.207 − 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.5 − 0.866i)10-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (0.207 + 0.978i)19-s + (−0.406 − 0.913i)20-s + (−0.5 − 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.994 + 0.104i)28-s + (−0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.994 − 0.104i)5-s + (0.207 − 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.5 − 0.866i)10-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (0.207 + 0.978i)19-s + (−0.406 − 0.913i)20-s + (−0.5 − 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.994 + 0.104i)28-s + (−0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.775 - 0.631i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8067040286 - 2.268985619i\)
\(L(\frac12)\) \(\approx\) \(0.8067040286 - 2.268985619i\)
\(L(1)\) \(\approx\) \(1.229015839 - 1.065302031i\)
\(L(1)\) \(\approx\) \(1.229015839 - 1.065302031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (0.994 - 0.104i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.406 - 0.913i)T \)
37 \( 1 + (0.207 - 0.978i)T \)
41 \( 1 + (-0.207 - 0.978i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−21.55569497525032851672156961283, −20.93032126467740704221515072993, −19.91111244069875832612551070427, −18.68166801262823046482817002520, −17.99781168860475957711092652221, −17.606718934533672751317364662842, −16.51984315500709687061938512630, −16.02265836313060939285947151904, −14.90339607006249194936527690076, −14.611477993212912779775110518, −13.57821508081194064790808022271, −13.11288887784662160009504102076, −12.10025697260052489825308898372, −11.52172367078546205215796797449, −10.23744419028211888283156019321, −9.26836933672117032929003062319, −8.78546797898278631709076857375, −7.71583670586562974279032090000, −6.87048146120663816551694356951, −5.99711153862052881336259515950, −5.3490829521368858619166583678, −4.778601433848266805597557850815, −3.31540818383659729876824765750, −2.68803302164841211671067987182, −1.51890206621335528474915954598, 0.782774715910610879622399179636, 1.7213878424790461263925160019, 2.52895868465402652261783070415, 3.765959351548923415633643205262, 4.29719668363334687996461439136, 5.55528485757791815588597730813, 5.915724694529617357897258303607, 7.06580419486974322584254258332, 8.13440098857848576661228204254, 9.24679910808707607365968566520, 10.07695330111127043350216866163, 10.424511958157352271100171180782, 11.33668947430511520337961365561, 12.3742301146832599840529245159, 12.91743224177295333594612700923, 13.88798784803506565777813546865, 14.19188037433606207762524034669, 14.96385048423724383334860314309, 16.22575467509616154046426774097, 16.99264985175108050189491525727, 17.71594697951239820539606325826, 18.61949379533276169246681589666, 19.2003018136807863508650580031, 20.40162225545771001399143970061, 20.6307091354764564247278471994

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