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Properties

Label 1-287-287.262-r1-0-0
Degree $1$
Conductor $287$
Sign $0.901 + 0.432i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.901 + 0.432i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.901 + 0.432i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.890025174 + 1.111603059i\)
\(L(\frac12)\) \(\approx\) \(4.890025174 + 1.111603059i\)
\(L(1)\) \(\approx\) \(2.539297257 + 0.2319821033i\)
\(L(1)\) \(\approx\) \(2.539297257 + 0.2319821033i\)

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

−25.05783787884325796773900904386, −24.46264303575087565805008623938, −23.4471659452847925058225283954, −22.86896709140383802139251566968, −21.36014839716780366934519631181, −21.0244898911210930310651440270, −20.042626355427873565991072924714, −18.80489378659187462319537881254, −17.78831935412334696084889180931, −17.098409509300139244613679906448, −15.80103760383082633126066796813, −14.9271656054055703959791148250, −13.76961230610960223097951019462, −13.36119297643971450608349312256, −12.65552694990876348130264811997, −11.483867028157598751194628452941, −10.2048932852951921849291593375, −8.7176769544387731696167871029, −7.92661662479145099514551941017, −6.799788496400428152552727501391, −5.87937524044158237503948538796, −5.05159993683003797323947647597, −3.28436057596123482933338571638, −2.52824407265363846706342013572, −1.16006186383857185744557505327, 1.56710233261138217609769281152, 2.72786175985030928813930951923, 3.58666265719627833414828807419, 4.93972951333648287827462376771, 5.59266950159702408868882409779, 6.8400723375181851959383389780, 8.37415464715188222451194089538, 9.69555460793197198720822374493, 10.30435811947970103277130075258, 11.1170158101463335409534617921, 12.495986955955903000188502009008, 13.5466916851184332017138457732, 14.159545231431599661389185927529, 14.96993270818252490630716335397, 16.006464412398238315339216254487, 16.76419536662888218345126882748, 18.36207548811442591527684886714, 19.11888726509829059329251446003, 20.49777293497935907051392986332, 21.042083962820620527054427466014, 21.420440722724506805810064925735, 22.61779330713305504998160591809, 23.20338799470255522826216804625, 24.52268122858507430743017729144, 25.47513606122329863032367383857

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