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Properties

Label 1-539-539.16-r0-0-0
Degree $1$
Conductor $539$
Sign $0.124 - 0.992i$
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  + (0.193 − 0.981i)2-s + (−0.447 − 0.894i)3-s + (−0.925 − 0.379i)4-s + (−0.842 + 0.538i)5-s + (−0.963 + 0.266i)6-s + (−0.550 + 0.834i)8-s + (−0.599 + 0.800i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (0.753 + 0.657i)13-s + (0.858 + 0.512i)15-s + (0.712 + 0.701i)16-s + (−0.999 − 0.0299i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (0.983 − 0.178i)20-s + ⋯
L(s)  = 1  + (0.193 − 0.981i)2-s + (−0.447 − 0.894i)3-s + (−0.925 − 0.379i)4-s + (−0.842 + 0.538i)5-s + (−0.963 + 0.266i)6-s + (−0.550 + 0.834i)8-s + (−0.599 + 0.800i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (0.753 + 0.657i)13-s + (0.858 + 0.512i)15-s + (0.712 + 0.701i)16-s + (−0.999 − 0.0299i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (0.983 − 0.178i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.124 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6463472326 - 0.5700915718i\)
\(L(\frac12)\) \(\approx\) \(0.6463472326 - 0.5700915718i\)
\(L(1)\) \(\approx\) \(0.6485837877 - 0.4542688133i\)
\(L(1)\) \(\approx\) \(0.6485837877 - 0.4542688133i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.193 - 0.981i)T \)
3 \( 1 + (-0.447 - 0.894i)T \)
5 \( 1 + (-0.842 + 0.538i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (-0.999 - 0.0299i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.971 + 0.237i)T \)
41 \( 1 + (-0.550 + 0.834i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.873 - 0.486i)T \)
53 \( 1 + (0.525 + 0.850i)T \)
59 \( 1 + (0.998 - 0.0598i)T \)
61 \( 1 + (0.525 - 0.850i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.983 + 0.178i)T \)
73 \( 1 + (-0.873 + 0.486i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.753 - 0.657i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−23.53194440727928958434084737053, −22.75390330588249775717670369479, −22.40324854304873869808290146723, −21.115575454422503926632138081460, −20.55173988259698260072781881587, −19.43509084360932145568875816740, −18.22508015189051502273204218431, −17.5150302011808112400071829725, −16.453898645872030857757120665582, −16.11251139108180037662215542912, −15.268315070870426093701945679135, −14.683194236704569532884081511132, −13.392943746385600008889444151193, −12.56098016907584707679314236980, −11.573839448772101913624025681370, −10.64294084641391302881413663296, −9.43565740348958806442151743848, −8.68026094418082810121922441271, −7.89170104026980233875960298783, −6.70816685093075917230682109970, −5.69135055645824916955544673952, −4.88887679666128055952920591921, −4.03738855581431056629934526926, −3.269833009472175379986960130549, −0.70995579904236574427479680169, 0.82574115449963113227426981988, 2.07182427437461609220121478005, 3.09432790129887909001215706556, 4.16830049937780604328072713631, 5.23402437458516965252930533306, 6.429315817783950167481669266, 7.300161973732655108185866691928, 8.39164705517299593942143140380, 9.303485412129543149552486936757, 10.72602520531141868784471108067, 11.40845436526441312382668269595, 11.69363675120778917446266046765, 12.9907323625090597215962052116, 13.445353862209565690646120580008, 14.482028124858455712228721148031, 15.45539694285067759066556704306, 16.633907408301923260267875280905, 17.747257790732347795886837978755, 18.42909365970917392441634904463, 19.01015330669002444990797436617, 19.81246668247791579567877670906, 20.469409481023956023663652644288, 21.85163851556532189276663289463, 22.32495936465353359517562826958, 23.28421504071603870022751869135

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