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Properties

Label 1-1287-1287.1229-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.843 - 0.537i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (−0.994 − 0.104i)7-s + (0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.207 − 0.978i)20-s + (−0.5 + 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.743 − 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (−0.994 − 0.104i)7-s + (0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.207 − 0.978i)20-s + (−0.5 + 0.866i)23-s + (0.104 + 0.994i)25-s + (−0.743 − 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.351045058 - 0.6861210879i\)
\(L(\frac12)\) \(\approx\) \(2.351045058 - 0.6861210879i\)
\(L(1)\) \(\approx\) \(1.443530191 + 0.1135779573i\)
\(L(1)\) \(\approx\) \(1.443530191 + 0.1135779573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
7 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.207 - 0.978i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.207 + 0.978i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.207 - 0.978i)T \)
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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

−20.995483281381330448272623043893, −20.03945176453979009622520413846, −19.49750939387576688491278847970, −18.859073464905165704793198110721, −18.164176407093093484732472866319, −16.580041634206195479956927144756, −16.25588689785692584856939400478, −15.33965239031572896522127374296, −14.66785902639015727551737817085, −14.09114950335349868675199884091, −12.931576697764289331989754442611, −12.504418176801639111280605271125, −11.70450604694259261226095010970, −10.808336428123204827308543944278, −10.25996636102000293815743840009, −9.30549016672156347915555961542, −8.03682092553906534813040067717, −7.13051025166460764152741874239, −6.45540944411290451311627968308, −5.72307102327188855826548257062, −4.54189713853541077057780501530, −3.74329215684756273531919232714, −3.0487675110342762278767374185, −2.26910165090898507721433161750, −0.78895667281235375537364730369, 0.42882205310531556376940844005, 1.84802228901438117274627081951, 3.04302110115246381646152238903, 3.87486966840238866780170426111, 4.38047661916105233813446871526, 5.66079278817103045198226531752, 6.079038202112475967304710084867, 7.34049789054192872627959170087, 7.77085611996397034511749325393, 8.8175623786563170241843698720, 9.77865508573091976213746804355, 10.82471259964265727169726954112, 11.734806486095578984334564758197, 12.37623035737421142470614845914, 13.02328157009949869825271206660, 13.62833447106662123108620451551, 14.75575792441296602358581075855, 15.35596964253027958336332377002, 16.08025596478143009998051291949, 16.75177580141728687992569033660, 17.218246466697706332976069212806, 18.75557105628747390410439833993, 19.37553544572079555404857898631, 20.13685176495663396740982157600, 20.76170893727040096102923219202

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