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Properties

Label 1-1287-1287.124-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.985 - 0.171i$
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  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (0.207 − 0.978i)5-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.994 − 0.104i)19-s + (−0.951 − 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (0.743 + 0.669i)28-s + (0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (0.207 − 0.978i)5-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.994 − 0.104i)19-s + (−0.951 − 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (0.743 + 0.669i)28-s + (0.913 − 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.985 - 0.171i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251589715 - 0.1082286665i\)
\(L(\frac12)\) \(\approx\) \(1.251589715 - 0.1082286665i\)
\(L(1)\) \(\approx\) \(0.7535762639 + 0.09465939797i\)
\(L(1)\) \(\approx\) \(0.7535762639 + 0.09465939797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
5 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 - T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 - T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.994 + 0.104i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.207 - 0.978i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−20.56388963115450319372645372529, −20.05859482889719858237693103533, −19.296199055468922038982352494876, −18.4884485089305488315965249606, −18.07189718008819561371415127795, −17.12966099484898759263391868700, −16.415183886640180699431535367812, −15.75693899871219856031076504914, −14.513859070140956765107871171633, −13.831625264760942205430961558023, −13.07676226879212089029165525198, −12.08959515321835995440957908491, −11.38080118409877787443930945774, −10.53767092194040964715660863257, −9.90003993536112901719601073906, −9.47990145399099244697718317749, −8.090467168119432251844043908753, −7.47983691319111666705628723111, −6.75643746442438068312069939057, −5.81900365940421297382856653258, −4.34181580645669696968620659661, −3.385985564093434008874020539737, −2.91673631983438691981645207843, −1.71612518121119062791599116512, −0.67336291117067903095943537775, 0.49146606306333720957533535421, 1.45452550457056977070874213017, 2.47537919379697122588235144006, 3.81538668784181942766565395703, 5.153968241981076276446008170733, 5.55806475304126978807592293834, 6.408191425634051050160014978625, 7.424969642586819539704601969051, 8.37036725696012675720609155209, 8.82369974872099793724101588473, 9.8832542467339292622111108681, 10.04707447092105332785686486480, 11.61515015485571852754472301865, 12.146167819800122735069112720437, 13.16034923760199705017668842567, 13.95983575185134501132030162485, 14.84229182422866095769722982553, 15.74910068935988281858838506906, 16.25709464516259442810283301104, 16.80479938669433621578145308649, 17.83367932877732045837413184771, 18.309586658806359592608589502349, 19.250121126606213067537171080787, 19.869429085788989624876783350841, 20.58717944701162711845679115101

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