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Properties

Label 1-539-539.24-r0-0-0
Degree $1$
Conductor $539$
Sign $0.729 - 0.683i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (0.999 + 0.0299i)2-s + (−0.420 − 0.907i)3-s + (0.998 + 0.0598i)4-s + (0.946 − 0.323i)5-s + (−0.393 − 0.919i)6-s + (0.995 + 0.0896i)8-s + (−0.646 + 0.762i)9-s + (0.955 − 0.294i)10-s + (−0.365 − 0.930i)12-s + (0.473 + 0.880i)13-s + (−0.691 − 0.722i)15-s + (0.992 + 0.119i)16-s + (0.887 − 0.460i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (0.963 − 0.266i)20-s + ⋯
L(s)  = 1  + (0.999 + 0.0299i)2-s + (−0.420 − 0.907i)3-s + (0.998 + 0.0598i)4-s + (0.946 − 0.323i)5-s + (−0.393 − 0.919i)6-s + (0.995 + 0.0896i)8-s + (−0.646 + 0.762i)9-s + (0.955 − 0.294i)10-s + (−0.365 − 0.930i)12-s + (0.473 + 0.880i)13-s + (−0.691 − 0.722i)15-s + (0.992 + 0.119i)16-s + (0.887 − 0.460i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (0.963 − 0.266i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.729 - 0.683i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.729 - 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.601075990 - 1.028191281i\)
\(L(\frac12)\) \(\approx\) \(2.601075990 - 1.028191281i\)
\(L(1)\) \(\approx\) \(1.950388224 - 0.5089054150i\)
\(L(1)\) \(\approx\) \(1.950388224 - 0.5089054150i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 + 0.0299i)T \)
3 \( 1 + (-0.420 - 0.907i)T \)
5 \( 1 + (0.946 - 0.323i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (0.887 - 0.460i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.936 - 0.351i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.772 + 0.635i)T \)
41 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.280 + 0.959i)T \)
53 \( 1 + (-0.842 + 0.538i)T \)
59 \( 1 + (-0.575 - 0.817i)T \)
61 \( 1 + (-0.842 - 0.538i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.963 - 0.266i)T \)
73 \( 1 + (-0.280 + 0.959i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.473 - 0.880i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−23.261114031654806265672402027217, −22.48999950215314954394228266046, −21.95234207963088376051336115833, −21.212027015843079611931422458795, −20.56350384534769610520893515789, −19.73695976948150647357709993928, −18.27875192367136931904402137538, −17.46926449528943293309654060868, −16.55106469459863711028977822996, −15.79943256296583301529855346238, −14.92056565223950663074169301034, −14.23933687134453290491792652197, −13.360046208894463851746492819918, −12.40043094174933413451347308625, −11.44550889045184744058578982990, −10.4727329346822669814434092105, −10.09555530147650341882569275303, −8.83076018078287527648776180273, −7.40439296413728986919904120054, −6.21400194041999014051271257146, −5.62314107128452771519139333220, −4.886760813610007433099792481762, −3.587737803016607160721148845785, −2.94333903688573560961055225350, −1.49063955239911303715311811796, 1.41035190598345380312600660068, 2.05089377920485648273326613594, 3.34090891624229075430831890181, 4.69700598492652305823148509132, 5.7258704366634824472131857450, 6.128504664821940131769198681737, 7.21293053338354852570174271309, 8.08742381614408719046342294518, 9.50596052063160114861327534890, 10.55533511354108236539871129289, 11.68121844225176955613657715570, 12.18909471627318255129987926729, 13.155646624201562582053973595837, 13.919953160126737876952335423249, 14.24552748595141224076019936260, 15.74927228836052327683244072187, 16.69064513582220074206451774992, 17.1435036816361969133503484466, 18.42917441685672303122102667499, 18.973417872039595308140470603214, 20.36853230396084141653651031647, 20.77475529515815663398102276998, 21.91182486690685222101804550291, 22.46253679136363498310554092214, 23.41293708822380051874226215125

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