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Properties

Label 1-1287-1287.1087-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.806 - 0.590i$
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  + (−0.994 − 0.104i)2-s + (0.978 + 0.207i)4-s + (0.994 − 0.104i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s − 10-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.587 − 0.809i)28-s + (0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.978 + 0.207i)4-s + (0.994 − 0.104i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s − 10-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.587 − 0.809i)28-s + (0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.806 - 0.590i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.806 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.709496264 - 0.5591326174i\)
\(L(\frac12)\) \(\approx\) \(1.709496264 - 0.5591326174i\)
\(L(1)\) \(\approx\) \(0.8910090000 - 0.1433693686i\)
\(L(1)\) \(\approx\) \(0.8910090000 - 0.1433693686i\)

Euler product

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

−20.73927254466884211727856069202, −20.0878732589583162934920601053, −19.0307333017122994998530379457, −18.71783024275000754865380197658, −17.81650871986017424638563695479, −17.19218442528407701460830829817, −16.47139152795913402401576809070, −15.66522703886811550113299053273, −14.97235333471008399452539580632, −14.03933309099473215376383614793, −13.11246632981607791539553687878, −12.26065691324498536440333451416, −11.46734667092377692266270432219, −10.431766816223791961784436357411, −9.803596084695415928316275591699, −9.24983054559994188205169653482, −8.47376021407560338421802313230, −7.41084033293955662998608481165, −6.607396986413409855853736717575, −5.81833090021644973109032237463, −5.24685625989633872308486479682, −3.40635634450489497471422635197, −2.69743013397528622510090481279, −1.73992587712023684738620714349, −0.769143207142049038804372474866, 0.712262243022441979196652465492, 1.321337077146202771679849036948, 2.67123780827707758469439913926, 3.19249070745184386618742999478, 4.64050692565201035314264955710, 5.82802850117963629093579677408, 6.49469910657261184638220106209, 7.28893015338685653469028066138, 8.15402133862607363975834766644, 9.19728492051465943675930596614, 9.80391129383419809544041021541, 10.26393204023975350955171433819, 11.16925743414118272774228041038, 12.195379535509410342007417295556, 12.89575819525760303917387438044, 13.84640472849070737225033148858, 14.52480484872852842967378535975, 15.72631394826466891452245225190, 16.49040571615413971465369831717, 16.83131152381782989803815885668, 17.82728851876615329649678559448, 18.29763274338927785497696103735, 19.21976990429603774074223728638, 19.869838367282431664034643471698, 20.80839992488398360769271066132

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