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Properties

Label 1-539-539.81-r0-0-0
Degree $1$
Conductor $539$
Sign $0.0168 - 0.999i$
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  + (−0.447 − 0.894i)2-s + (−0.337 − 0.941i)3-s + (−0.599 + 0.800i)4-s + (−0.925 − 0.379i)5-s + (−0.691 + 0.722i)6-s + (0.983 + 0.178i)8-s + (−0.772 + 0.635i)9-s + (0.0747 + 0.997i)10-s + (0.955 + 0.294i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.280 − 0.959i)16-s + (0.420 + 0.907i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.858 − 0.512i)20-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)2-s + (−0.337 − 0.941i)3-s + (−0.599 + 0.800i)4-s + (−0.925 − 0.379i)5-s + (−0.691 + 0.722i)6-s + (0.983 + 0.178i)8-s + (−0.772 + 0.635i)9-s + (0.0747 + 0.997i)10-s + (0.955 + 0.294i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.280 − 0.959i)16-s + (0.420 + 0.907i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.858 − 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0168 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0168 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.0168 - 0.999i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.0168 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4471659441 - 0.4547565669i\)
\(L(\frac12)\) \(\approx\) \(0.4471659441 - 0.4547565669i\)
\(L(1)\) \(\approx\) \(0.4989716466 - 0.3577577521i\)
\(L(1)\) \(\approx\) \(0.4989716466 - 0.3577577521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.447 - 0.894i)T \)
3 \( 1 + (-0.337 - 0.941i)T \)
5 \( 1 + (-0.925 - 0.379i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (0.420 + 0.907i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.946 + 0.323i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.887 + 0.460i)T \)
53 \( 1 + (0.575 + 0.817i)T \)
59 \( 1 + (-0.646 - 0.762i)T \)
61 \( 1 + (0.575 - 0.817i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (0.887 - 0.460i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.550 - 0.834i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−23.51054415610245560383540658140, −22.7315618965369956314342919915, −22.47634363494402045535404359738, −21.198803849417925777915765325662, −20.039843875214830866501994991399, −19.51601456942399670747863472483, −18.29819304496747083418876506680, −17.6863944927261820361620480594, −16.68546862998646722819464007001, −15.81653781320530703504786440100, −15.54887384267153553799896220283, −14.53893761502513751827622862165, −13.868251119810785230210547280469, −12.21774666164457022719401606970, −11.4627569946192648984275964732, −10.287455863838049345259319959189, −9.87461311415244198221775153724, −8.687513734697223589937128250468, −7.80502881249926561929671322596, −7.003441543710776672459459470280, −5.76098588646465868926166081704, −5.02260948527766744979167861400, −4.016059413531729267076686891182, −2.96594382385030541415036134673, −0.69337979250541111978142710489, 0.74830984989947716607007391329, 1.83520995936920387516714411768, 2.98833382790909052941558697639, 4.172191130024461572961596851393, 5.12962236462729556165647872035, 6.635206585503896288319221894849, 7.617697478373045326915834741185, 8.2513552710906891088989899200, 9.17435649065189687300219886607, 10.407524251674339836418778893623, 11.33421421678872074438168156927, 12.13914672288192292217186192298, 12.42393730612880733008538081081, 13.56302423740274309662463470466, 14.38398084860646219704203929656, 15.87132888416189595527232423768, 16.78441386962420525363831656151, 17.381431721334498617646560788, 18.48451059796163462090024858629, 19.01263624242952732379884096643, 19.78195726217908544815122756838, 20.33303330086310478211127801666, 21.54230785678948756126732716502, 22.36854377183439480343099984233, 23.19516709396593558382821455920

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