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Properties

Label 1-287-287.139-r1-0-0
Degree $1$
Conductor $287$
Sign $0.747 - 0.664i$
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  + (−0.809 − 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + 9-s + (−0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + 9-s + (−0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.747 - 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7403137267 - 0.2815775804i\)
\(L(\frac12)\) \(\approx\) \(0.7403137267 - 0.2815775804i\)
\(L(1)\) \(\approx\) \(0.5403646818 - 0.1888344403i\)
\(L(1)\) \(\approx\) \(0.5403646818 - 0.1888344403i\)

Euler product

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

−25.42405318329474495004552493962, −24.63921599080583671085510696110, −23.44095206507769602447668249064, −22.90271097510693295986859800304, −22.24321774065259108786551013173, −20.723687761244013862712454813309, −19.74633629455089451173815144417, −18.54684625077674350297171013628, −18.08465987430914087386417561764, −17.39836983086086157538864701043, −16.14474437003146977576810596038, −15.60866496337695391918721157800, −14.65687883789732812633114024703, −13.46732405324895107171355548735, −11.85113722772837580802099241652, −11.28601993628188027759691950899, −10.19276081339504712215275782229, −9.607753293958138667014011972305, −7.90482500458937474338327881260, −7.17995289449162544018959343724, −6.27824776825914405866771847136, −5.38082760213244758520544545984, −3.953780212521661583547294229637, −2.0950528740233579178912422719, −0.60492323079530022543601517354, 0.73781797368920213217216597281, 1.5745116105178523514966784387, 3.549244562532489619719124871547, 4.53240496192720867990506691405, 5.91935248082878446040752971974, 6.990499459779763876085997268966, 8.35470824737814525075597147986, 8.99977402056757934703701029675, 10.26493881189267657570694775536, 11.19519913273287924004150774588, 11.8723058390625155556945002719, 12.72788134286135511767811291914, 13.67494613446637932349645376558, 15.69921893532113326102668025025, 16.30893940266132430964617654425, 16.94230334486889619034753097098, 17.90597145411182706096868193096, 18.752264740858688093284742003460, 19.70109071832958170346140830165, 20.600356757274193485158043682952, 21.6385138211062263638894062360, 22.11252593166635642162233028054, 23.64340281980727558692947952517, 24.14054459566906432900443954334, 25.20639305282964064572849788207

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