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Properties

Label 1-287-287.256-r0-0-0
Degree $1$
Conductor $287$
Sign $0.881 + 0.472i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.978 − 0.207i)2-s + (−0.5 − 0.866i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)10-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.5 − 0.866i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)10-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)18-s + (0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4490484099 + 0.1127449980i\)
\(L(\frac12)\) \(\approx\) \(0.4490484099 + 0.1127449980i\)
\(L(1)\) \(\approx\) \(0.5254654504 - 0.1030408968i\)
\(L(1)\) \(\approx\) \(0.5254654504 - 0.1030408968i\)

Euler product

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

−25.99447943755234785274639332176, −24.67613782457836230285352963727, −23.70915670469561919975041183476, −22.60386450122649503087381117043, −21.95776049984585702541186212841, −20.78340286985148294534488031131, −20.03034266482592828355361915928, −18.82053894785717106878962485779, −18.097088019215164936096150176350, −17.328651509760046315056550462709, −16.17399452328006846943310117214, −15.62195331577692738409083655189, −14.78926606609016214976220848278, −13.630451750869049681576964472220, −11.62706750999216769695988024085, −11.29658405751287862346049048788, −10.281229787275430639082319171445, −9.599745978091407145538281025634, −8.39825498333579939370223870549, −7.31174578842345277841935127942, −6.16166047008085357253790403447, −5.41139266439424855785767052773, −3.576301714623484936406286528249, −2.64659213701539717383198759363, −0.47110724577518440581620685657, 1.370924805815401430230937936372, 2.015933512344798463095234737702, 3.96550157791811353422626622538, 5.46882768752753138148858520640, 6.58031115276743476487249802779, 7.59547130906234327791785616840, 8.407754258609150581552113261, 9.42454236164761262739015687094, 10.546211603232889401108244363556, 11.71407036381062172235816526319, 12.33662773643095107238617410710, 13.08670099724653253872482469413, 14.534574674650371927545214639352, 16.086686431661105327612113515148, 16.53441254185545245497806666899, 17.57907096249770748955282397176, 18.14245979332051601897983576686, 19.23967586186374684852613636766, 19.93128976442205640143463364118, 20.78730723241902255181145571519, 21.83155399153556950530203589080, 23.187276460363787876247057392534, 24.03276552842467072401965178585, 24.705205611405307733918187871379, 25.57186567667912486425784382982

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