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Properties

Label 1-287-287.187-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.780 + 0.625i$
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 + (−0.5 + 0.866i)3-s + (0.669 + 0.743i)4-s + (0.978 − 0.207i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)9-s + (0.978 + 0.207i)10-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)12-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.5 + 0.866i)3-s + (0.669 + 0.743i)4-s + (0.978 − 0.207i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)9-s + (0.978 + 0.207i)10-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)12-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.780 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071525682 + 3.048674812i\)
\(L(\frac12)\) \(\approx\) \(1.071525682 + 3.048674812i\)
\(L(1)\) \(\approx\) \(1.415667753 + 1.146989830i\)
\(L(1)\) \(\approx\) \(1.415667753 + 1.146989830i\)

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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−24.70422185769342398170469580460, −24.18168019425149546321939751355, −22.91186186058462992726107747758, −22.25845002772443040517698803670, −21.71550832278046503172453897862, −20.44257858565479377155195348440, −19.51691471342776663620645006638, −18.77543809343790754221376663647, −17.44824655535836270758986702103, −17.05857462601079532875799774908, −15.46816100908806757636156566356, −14.46093348790277282330147585587, −13.55633837580880587646615659530, −12.98937942789612526168381323879, −11.96278758628046356632821347093, −11.09185805743672967121319521299, −10.172990046580571910811274089183, −8.89522101770734071487561334928, −7.11122055672910231415211790344, −6.50069074743337884247230586492, −5.56179472701728025183962049832, −4.57725365361268799546715826970, −2.83155228056241806815744224919, −1.997101712546951092797175211021, −0.73696752126241850844865825481, 1.74150505001825472636374132338, 3.20012970871258110279394786244, 4.45132823243430112417814708334, 5.12166285856582438841058868675, 6.23346930234705588431322396764, 6.93231892673882587121502101056, 8.74157782841704511675971308908, 9.59299760306246989653511133375, 10.74348176076420454841306724411, 11.79335500171311389623878443746, 12.5925007754771949000850003333, 13.85870111357982421006476954068, 14.5506331495606612667543726511, 15.423138931038826085078546974473, 16.62899528287895385057780784636, 17.01858100713520516117553742046, 17.874111626315816675289601490765, 19.647680101251933833847741172804, 20.70693347664920889623284170558, 21.32823447971064437449259338900, 22.213468038294221892451207227, 22.60062092235144235526378083597, 23.82488269371093761985401801861, 24.74405700023290824891219711717, 25.45282780649270187360947343354

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