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Properties

Label 1-287-287.144-r0-0-0
Degree $1$
Conductor $287$
Sign $0.891 - 0.452i$
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  + (−0.913 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.891 - 0.452i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273646645 - 0.3050269496i\)
\(L(\frac12)\) \(\approx\) \(1.273646645 - 0.3050269496i\)
\(L(1)\) \(\approx\) \(1.098904945 - 0.1028864866i\)
\(L(1)\) \(\approx\) \(1.098904945 - 0.1028864866i\)

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

−25.67227619972409126082398724542, −25.260328893917922073162729968927, −24.181614606815661640233556708755, −22.57681013259533547919977653640, −21.497258029157110226046110548776, −20.84643644422310764305837855637, −20.39501126125427662590424756920, −19.1768681531425269127424757464, −18.46745195803313673500788382713, −17.39706321301027409226035954119, −16.57602436354628971613106852826, −15.618908094099872395125200352018, −14.57374025999165225690043280240, −13.434060222635684668956921365673, −12.654289563465236315740225579288, −11.19100704346685957034457169658, −10.27051122922228586505812843924, −9.31573295691553448927376428544, −8.99194013314929649537685195041, −7.65090041704365769424034244586, −6.68103520751217382484827096930, −5.01781373911263187780701302551, −3.722817283750957616818688309790, −2.395685996058556063366271976462, −1.68994707955190216774568024961, 1.19145031883221335304977294148, 2.26108038904212968778696430423, 3.39823426364772655965614891392, 5.586535281164050562041562740579, 6.32475789008974016666475972590, 7.399930925180864247465522657097, 8.55179463678715073762396149168, 8.976762206615196501026697421681, 10.3067154892109456929875095308, 10.91106560398447491266412052456, 12.68929630399162938835755705125, 13.493778306373721391741387523853, 14.57308750888966824782491761009, 15.160233739907048260003588582601, 16.479979547944484693973970583509, 17.373262364427993916499792810255, 18.3142359212575031603702423113, 18.879272123050480496039353130504, 19.78670933896115706070129472572, 20.85253034610559754814356762724, 21.43883004976650227193094348349, 23.043315249200591585593802379698, 24.1396206716691567122801178534, 24.73525341793492639024783578230, 25.757035414381588741977457163858

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