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Properties

Label 1-539-539.212-r0-0-0
Degree $1$
Conductor $539$
Sign $0.714 + 0.700i$
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  + (0.420 + 0.907i)2-s + (0.712 − 0.701i)3-s + (−0.646 + 0.762i)4-s + (0.998 + 0.0598i)5-s + (0.936 + 0.351i)6-s + (−0.963 − 0.266i)8-s + (0.0149 − 0.999i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (−0.163 − 0.986i)16-s + (0.791 + 0.611i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.691 + 0.722i)20-s + ⋯
L(s)  = 1  + (0.420 + 0.907i)2-s + (0.712 − 0.701i)3-s + (−0.646 + 0.762i)4-s + (0.998 + 0.0598i)5-s + (0.936 + 0.351i)6-s + (−0.963 − 0.266i)8-s + (0.0149 − 0.999i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (−0.163 − 0.986i)16-s + (0.791 + 0.611i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.691 + 0.722i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.714 + 0.700i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.714 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.204498408 + 0.9003971527i\)
\(L(\frac12)\) \(\approx\) \(2.204498408 + 0.9003971527i\)
\(L(1)\) \(\approx\) \(1.670692359 + 0.5167707321i\)
\(L(1)\) \(\approx\) \(1.670692359 + 0.5167707321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.420 + 0.907i)T \)
3 \( 1 + (0.712 - 0.701i)T \)
5 \( 1 + (0.998 + 0.0598i)T \)
13 \( 1 + (-0.995 - 0.0896i)T \)
17 \( 1 + (0.791 + 0.611i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.473 + 0.880i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.525 - 0.850i)T \)
41 \( 1 + (-0.963 - 0.266i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.193 - 0.981i)T \)
53 \( 1 + (-0.925 - 0.379i)T \)
59 \( 1 + (0.251 - 0.967i)T \)
61 \( 1 + (-0.925 + 0.379i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (0.193 + 0.981i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.995 + 0.0896i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−22.87501435936906095403610954479, −22.22943038472562439565016884427, −21.543413219085637467972972976990, −20.80358757303859257260719760581, −20.30952454150580326915288270995, −19.28712772535765051120435585633, −18.586926549250296063374474274279, −17.47785140933461426366651251309, −16.60314648106583668607345798747, −15.35275317904316018042821372506, −14.55714651436969038470702768786, −13.875438631646977173132269151277, −13.24450779257105102953689635133, −12.16292582450740479083171209833, −11.14505029825545045140349991303, −10.03033419407898434439619341376, −9.70986333665812774116625467393, −8.918447138765600534183645884299, −7.64545784620822284856783517589, −6.140645793449963189397927842978, −5.021525264061353829317021592337, −4.5345216953729951813672267697, −2.949045837122365498454016237388, −2.64772893688775606821428589494, −1.277298800688454953525146095338, 1.34168266606770987498607959686, 2.69399562219821088200218277646, 3.513469656989423252898045931597, 5.03257943779167021281081561386, 5.80357400019591371098986328010, 6.8598786654886218032744145779, 7.4778022842680929328165706444, 8.51835936075758632519149226199, 9.350523802420450605146017093817, 10.179710195037110987332688535, 11.98790836478514285214589116813, 12.66035760342473844457505700338, 13.456818545819151978660480274808, 14.29343916319830460307349701246, 14.65277240852954234003815474036, 15.759653660460325757076226476222, 16.94544766965244471540205120015, 17.51889976349662402655516062200, 18.32448095763346728599251138659, 19.1487128852983591190974787497, 20.28272110642863271189128453241, 21.23320290753120759369198742345, 21.83404146090876768333236787654, 22.87118795619341275198378203559, 23.739270095531752593195718611591

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