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Properties

Label 1-287-287.166-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.970 + 0.241i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (−0.913 − 0.406i)2-s + (−0.866 − 0.5i)3-s + (0.669 + 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)9-s + (0.978 + 0.207i)10-s + (−0.207 + 0.978i)11-s + (−0.207 − 0.978i)12-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.104 − 0.994i)18-s + (0.994 + 0.104i)19-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.866 − 0.5i)3-s + (0.669 + 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)9-s + (0.978 + 0.207i)10-s + (−0.207 + 0.978i)11-s + (−0.207 − 0.978i)12-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.104 − 0.994i)18-s + (0.994 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.970 + 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002864186639 + 0.02339937714i\)
\(L(\frac12)\) \(\approx\) \(0.002864186639 + 0.02339937714i\)
\(L(1)\) \(\approx\) \(0.4049508516 - 0.05685805404i\)
\(L(1)\) \(\approx\) \(0.4049508516 - 0.05685805404i\)

Euler product

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

−24.48900892443091076868120237061, −24.28453657183729813080564724303, −23.198621800287557804375752423480, −22.40376554597678405319855848050, −21.100615288052277993549071949184, −20.303188609267917472674962402107, −19.10852034778671019418146683979, −18.56254053746731586143542804359, −17.38154276618603600357539243839, −16.5018872584743219488479065776, −16.01276340597285514792109664388, −15.20710364585994349688541288358, −14.030465998341324349995090229888, −12.29447310472508396360978204937, −11.405289829084192495312220674376, −10.93049102635493335790411845552, −9.59025984184214447178165462512, −8.85736051536676412942317924902, −7.54944744700602258384489374885, −6.76926253030326642359598328250, −5.49298760041041503147942780633, −4.58928913542086039389291594918, −3.043519845303537517959957338484, −0.97879870357320184678378780561, −0.015227620191580599264135378466, 1.24494172680116367584746915809, 2.646167394980677064769109511375, 4.05518923024637345398124813133, 5.451364101253770172858278258836, 7.02501313864653294848702971126, 7.42824414897219026164544243856, 8.45475502079828977485178309420, 9.97701475893472436928303979794, 10.68599699008120979969168388690, 11.698945143683470685091296452105, 12.313120624152482050292812830884, 13.151632867522680773182020357910, 15.108567521292401917912269255131, 15.724308894817933129351600145070, 16.913396191503090491245312371434, 17.56251901759729713499038679310, 18.41727596772736799632950388610, 19.294071890772364809390478449313, 19.9502756545467515807415257644, 21.02476674199220375630391849098, 22.35608895672876369421947473726, 22.83193326752764644813367155696, 24.0416312599626850170232041879, 24.7689107846372200309963528839, 25.87152804884854909377761975581

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