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Properties

Label 1-539-539.138-r0-0-0
Degree $1$
Conductor $539$
Sign $0.996 - 0.0805i$
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  + (0.646 + 0.762i)2-s + (−0.0149 − 0.999i)3-s + (−0.163 + 0.986i)4-s + (−0.992 + 0.119i)5-s + (0.753 − 0.657i)6-s + (−0.858 + 0.512i)8-s + (−0.999 + 0.0299i)9-s + (−0.733 − 0.680i)10-s + (0.988 + 0.149i)12-s + (0.983 − 0.178i)13-s + (0.134 + 0.990i)15-s + (−0.946 − 0.323i)16-s + (0.251 − 0.967i)17-s + (−0.669 − 0.743i)18-s + (0.669 − 0.743i)19-s + (0.0448 − 0.998i)20-s + ⋯
L(s)  = 1  + (0.646 + 0.762i)2-s + (−0.0149 − 0.999i)3-s + (−0.163 + 0.986i)4-s + (−0.992 + 0.119i)5-s + (0.753 − 0.657i)6-s + (−0.858 + 0.512i)8-s + (−0.999 + 0.0299i)9-s + (−0.733 − 0.680i)10-s + (0.988 + 0.149i)12-s + (0.983 − 0.178i)13-s + (0.134 + 0.990i)15-s + (−0.946 − 0.323i)16-s + (0.251 − 0.967i)17-s + (−0.669 − 0.743i)18-s + (0.669 − 0.743i)19-s + (0.0448 − 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.996 - 0.0805i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.996 - 0.0805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487872069 - 0.05999142923i\)
\(L(\frac12)\) \(\approx\) \(1.487872069 - 0.05999142923i\)
\(L(1)\) \(\approx\) \(1.218087508 + 0.1212649895i\)
\(L(1)\) \(\approx\) \(1.218087508 + 0.1212649895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.646 + 0.762i)T \)
3 \( 1 + (-0.0149 - 0.999i)T \)
5 \( 1 + (-0.992 + 0.119i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (0.251 - 0.967i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.550 + 0.834i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.447 + 0.894i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.925 - 0.379i)T \)
53 \( 1 + (0.712 - 0.701i)T \)
59 \( 1 + (0.873 - 0.486i)T \)
61 \( 1 + (0.712 + 0.701i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (-0.925 - 0.379i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.983 + 0.178i)T \)
89 \( 1 + (-0.826 - 0.563i)T \)
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

−23.07393487164533952955076936125, −22.73953413202603210985758775155, −21.58516822625458849198705476258, −21.003984753489870538669431846939, −20.2781917139363714139290734827, −19.46645003715074962621108291230, −18.79228650289125232125984308325, −17.536814091591718876137178461440, −16.300699991029009627565458909, −15.673619582217002740091263374469, −14.94583739131225178743615380070, −14.1304690612179461574056137521, −13.0965055098692542030433460599, −12.02885363474085550404173513871, −11.35204979861187709064939380852, −10.6804903544727935059225583225, −9.71849489434627710723122608650, −8.80589814854958725001813423546, −7.79056734085350120352508990222, −6.15466647743007855517534810338, −5.39125610008956476917515075276, −4.0865941797176346477854262388, −3.87133167834926125936316690428, −2.77054189031928683562364801113, −1.1233331002239607325241093989, 0.80153059717474606617364823357, 2.71472523319819830516681802554, 3.44957768632194396951056844350, 4.735005581440933098846281076064, 5.674782202273638645073486691563, 6.97736204116462296869724874021, 7.13043089369718329119231975047, 8.38832674699770966719942921091, 8.81402981790441626521366957450, 10.80010026346591318937217704909, 11.748740025310385237233570787840, 12.289098706786228419897368565657, 13.26776699591246104479341493211, 13.98769524037840938599947679704, 14.81452874587102350266111606310, 15.87452681007484930527435756283, 16.35667447363382828281235246252, 17.59252155819007611755681578325, 18.266468177570155726398864054231, 19.052219060032329899155552682493, 20.1787340332164074192005967091, 20.781121790280012052522378607531, 22.25896519700277493770710933724, 22.77896294790123624305517816558, 23.56978792391885496815006975260

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