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Properties

Label 1-2667-2667.2036-r0-0-0
Degree $1$
Conductor $2667$
Sign $0.569 + 0.821i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.222 + 0.974i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 19-s + 20-s + 22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.222 + 0.974i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 19-s + 20-s + 22-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (2036, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5910706448 + 0.3093672684i\)
\(L(\frac12)\) \(\approx\) \(0.5910706448 + 0.3093672684i\)
\(L(1)\) \(\approx\) \(0.7722100567 - 0.2529056427i\)
\(L(1)\) \(\approx\) \(0.7722100567 - 0.2529056427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−18.98756122019902567417543229988, −18.594452823266625262476187530514, −17.59328052383800927507287392200, −16.87207030741466810152971327356, −16.30461084816757392245620758528, −15.692095315327160950171978673597, −15.16462356273144385060826766683, −14.32357048857233614031500177323, −13.40561609124675489380873060914, −13.16581823666067078551944867161, −12.08364021611827745127062653317, −11.41387277226058393665240857408, −10.74041023485974743291385521154, −9.37883499493508249829743756420, −8.80930396966496173328983698122, −8.330695570809480515745659919957, −7.50174891926546315206436837605, −6.71403888416874474968011349196, −6.0298147023941782833586576579, −5.1828619947833952037784699356, −4.27334187114498186738033594692, −3.817897482707566554895977757330, −2.921500276287535583782434397209, −1.35729449831460645177448698059, −0.24135500756300767257004506077, 0.99734720737704224533666945922, 2.12298021036743795136009511408, 2.78440086227021660419935629617, 3.875691620051628985467272813049, 4.22387688201117815398438719896, 5.04128446996223553997983702486, 6.30152942956034796510013071484, 6.77798717764938772006310332108, 8.24373177137096314256711773810, 8.317954302411526568221240532494, 9.50320820183497062606731498112, 10.239767669979005670608236571944, 11.00011921672746161330229372273, 11.37188767732686797207079932898, 12.31064057217177842016203426245, 12.85787561854093494878109315057, 13.49408441036405710840192782448, 14.59984884088488632254495442929, 15.004938296456516275954197379744, 15.61123090255157395846048613730, 16.68489812972379053992947975328, 17.60766820776486460467475366097, 18.157108180401285706252730622193, 18.87958106291975864933023994669, 19.52112359531801021915081417417

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