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Properties

Label 1-1287-1287.1039-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.943 - 0.331i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + 10-s + (0.669 − 0.743i)14-s + (0.913 − 0.406i)16-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)28-s + (0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + 10-s + (0.669 − 0.743i)14-s + (0.913 − 0.406i)16-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)28-s + (0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02107581284 - 0.1234401915i\)
\(L(\frac12)\) \(\approx\) \(0.02107581284 - 0.1234401915i\)
\(L(1)\) \(\approx\) \(0.7198573494 + 0.1061793655i\)
\(L(1)\) \(\approx\) \(0.7198573494 + 0.1061793655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 - T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−21.54500151092352514890544708126, −20.63488678455697377540919135107, −19.55292576145971944573173370054, −19.25757084622004081279060244001, −18.43094302056934948152920733249, −17.87274000743034038222364076976, −17.033014316901339423314177183429, −15.75489934781732584801259953321, −15.134117819354868493650249132979, −14.26454697586066451241645403371, −13.590954055932329901294071211660, −12.69921864677491542811115682483, −12.03477648362734566351605166251, −11.2359812008938061411697427956, −10.419601103834693395522853866724, −9.86126462451223634247478590431, −8.97308472772985576738095725590, −8.1673295967652210783670196423, −6.95801567750179436981977673367, −6.016076855620644621521216390363, −5.383754401927260400163894471217, −3.92601325390237356777058538841, −3.47741280613978973635723335310, −2.36095963891512487864477625689, −1.85367280993770405099296689748, 0.05091409415438603780593517116, 1.13234857986796218786127284304, 2.75227434884863317454797727748, 3.98435809203707920552227363174, 4.60315724265168773332899870281, 5.36849521418820711654182230118, 6.503983864486882868898334566454, 6.94169623841260210188521286120, 8.049484200211986627235385795667, 8.74792429175825135493327345506, 9.478640302687827752008916193487, 10.20983015030161126709507474807, 11.40784529384986324125729431651, 12.574379975269547427643667340839, 13.06637889084666441702740007399, 13.69866580627519499007657144646, 14.48724355982634670036535994136, 15.6077156489645947683956746624, 16.06711528806142818785264462609, 16.77828912038573705097358488430, 17.36322191423842963375445820305, 18.14032649130458001588398386535, 19.084049511313446057567038988744, 20.0700680967511915294107233746, 20.44989409959479639775594990618

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