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Properties

Label 1-539-539.163-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.215 + 0.976i$
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  + (0.992 − 0.119i)2-s + (−0.163 + 0.986i)3-s + (0.971 − 0.237i)4-s + (0.251 + 0.967i)5-s + (−0.0448 + 0.998i)6-s + (0.936 − 0.351i)8-s + (−0.946 − 0.323i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (−0.393 + 0.919i)13-s + (−0.995 + 0.0896i)15-s + (0.887 − 0.460i)16-s + (−0.337 + 0.941i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.473 + 0.880i)20-s + ⋯
L(s)  = 1  + (0.992 − 0.119i)2-s + (−0.163 + 0.986i)3-s + (0.971 − 0.237i)4-s + (0.251 + 0.967i)5-s + (−0.0448 + 0.998i)6-s + (0.936 − 0.351i)8-s + (−0.946 − 0.323i)9-s + (0.365 + 0.930i)10-s + (0.0747 + 0.997i)12-s + (−0.393 + 0.919i)13-s + (−0.995 + 0.0896i)15-s + (0.887 − 0.460i)16-s + (−0.337 + 0.941i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.473 + 0.880i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485527644 + 1.848209983i\)
\(L(\frac12)\) \(\approx\) \(1.485527644 + 1.848209983i\)
\(L(1)\) \(\approx\) \(1.592374820 + 0.8334280674i\)
\(L(1)\) \(\approx\) \(1.592374820 + 0.8334280674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.992 - 0.119i)T \)
3 \( 1 + (-0.163 + 0.986i)T \)
5 \( 1 + (0.251 + 0.967i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (-0.337 + 0.941i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.925 + 0.379i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.420 + 0.907i)T \)
53 \( 1 + (-0.646 + 0.762i)T \)
59 \( 1 + (-0.772 + 0.635i)T \)
61 \( 1 + (-0.646 - 0.762i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.420 - 0.907i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.393 - 0.919i)T \)
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

−23.15258184309199089916146842847, −22.66442533234069114968668310071, −21.55051479781449479809664917834, −20.69086960164126176238840754867, −19.95913569845150518861939641350, −19.27574304095268119970179152296, −17.92010412969172204959964296295, −17.21444066994840336102277088291, −16.43419460255893437708239276594, −15.48562286492284509066045018478, −14.40238244872252405039522973203, −13.628432457397751998829085000065, −12.70473631023012599576806756137, −12.52629589001881199692626639401, −11.422475052704277023575513031479, −10.48612121991653717240374221476, −8.94705802607503908875263297302, −8.0609417759743596783245128836, −7.10284856185594563011509092627, −6.239856124925185979883690598698, −5.24518988324276086308009520182, −4.6428955342420761361380452701, −3.04455807354173817567323448970, −2.14992254027206961772294245580, −0.90432537738636938945619195866, 1.98703729280403468496694605887, 2.92859811965310105065831600850, 3.96089315572182877129823567729, 4.62939700927927413116764205984, 5.926087552863886781573207564253, 6.428972452289673003538885935894, 7.606860784643818685667642168244, 9.069383896742959254551301349689, 10.133748536578653740930228440321, 10.785984708553502171202114784987, 11.477123766632690622872794519983, 12.439137877036487016193861165681, 13.712025276489092115028463377141, 14.31049067202713382165392921922, 15.233734024591157163794728019934, 15.547016703375294192596765642274, 16.92417056558989505769653710967, 17.34975131667929111630326433840, 19.06550713388291106755431503723, 19.43254122394036948035113035951, 20.82876459753850146159071235163, 21.305666076328830028456790617287, 21.96532366738376703075534618332, 22.69476876579494615744877766896, 23.34509876587191722405589913226

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