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Properties

Label 1-287-287.4-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.649 + 0.759i$
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  + (0.669 + 0.743i)2-s + (0.5 + 0.866i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.913 + 0.406i)10-s + (−0.913 − 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.5 + 0.866i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.913 + 0.406i)10-s + (−0.913 − 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.649 + 0.759i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.649 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8839342497 + 1.919159538i\)
\(L(\frac12)\) \(\approx\) \(0.8839342497 + 1.919159538i\)
\(L(1)\) \(\approx\) \(1.262233311 + 1.195524662i\)
\(L(1)\) \(\approx\) \(1.262233311 + 1.195524662i\)

Euler product

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

−24.83496153027059301136227027659, −24.47376333289036279515813134186, −23.19108091379998872918411339323, −22.57666022251195553574639533471, −21.502177661128166627485676883611, −20.60270089870553527706986796634, −19.95531043528499793024173265038, −18.87854050152466575676686381575, −18.08607267586970793099664307957, −17.486714433285047095857881576341, −15.50403865998043443585205630702, −14.77802647936692288942855162433, −13.76205906076987226045622014563, −13.14867725837623675121543113315, −12.470009423533524536398231362612, −11.19069310744547139699123145575, −10.19647324257725517762648397262, −9.33363701518258684613800042967, −7.96327412666126029118262193524, −6.69452769481151933022956800659, −5.83834873942922270561513872039, −4.67400917245354866989806455809, −2.83784397169118367733357233295, −2.57283952002745713123881069461, −1.14353016963895250382010365361, 2.28939254213424148055098466034, 3.302439165577102391463857047569, 4.74230090334563815121347836799, 5.19730624660034528264852162051, 6.44289431959701919196637752713, 7.734147123716402604067053373776, 8.85143918795284469785001029469, 9.49201500441869434974214576798, 10.781527798887485614630199530839, 12.019675938377822444483676028453, 13.4882023659635902460567336467, 13.69823478621185587173236864206, 14.75861345554653134472532046199, 15.842133312538434316115319997673, 16.358693188381459928854350416412, 17.33171192501618672975408980500, 18.30983717514402876324615105182, 19.83649465950474884843924735654, 20.86622404571755933641880939557, 21.43422601786870021937103245389, 22.03782189576876372545230072872, 23.13118891065472388735888600358, 24.268216187130877525005771276641, 24.933060398819088902323072877472, 25.82063208815583580553292116791

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