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Properties

Label 1-539-539.179-r0-0-0
Degree $1$
Conductor $539$
Sign $0.929 - 0.369i$
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  + (−0.163 + 0.986i)2-s + (−0.999 + 0.0299i)3-s + (−0.946 − 0.323i)4-s + (0.971 − 0.237i)5-s + (0.134 − 0.990i)6-s + (0.473 − 0.880i)8-s + (0.998 − 0.0598i)9-s + (0.0747 + 0.997i)10-s + (0.955 + 0.294i)12-s + (0.936 − 0.351i)13-s + (−0.963 + 0.266i)15-s + (0.791 + 0.611i)16-s + (−0.873 − 0.486i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.995 − 0.0896i)20-s + ⋯
L(s)  = 1  + (−0.163 + 0.986i)2-s + (−0.999 + 0.0299i)3-s + (−0.946 − 0.323i)4-s + (0.971 − 0.237i)5-s + (0.134 − 0.990i)6-s + (0.473 − 0.880i)8-s + (0.998 − 0.0598i)9-s + (0.0747 + 0.997i)10-s + (0.955 + 0.294i)12-s + (0.936 − 0.351i)13-s + (−0.963 + 0.266i)15-s + (0.791 + 0.611i)16-s + (−0.873 − 0.486i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.995 − 0.0896i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.929 - 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7914397764 - 0.1516100431i\)
\(L(\frac12)\) \(\approx\) \(0.7914397764 - 0.1516100431i\)
\(L(1)\) \(\approx\) \(0.7414432148 + 0.1547169611i\)
\(L(1)\) \(\approx\) \(0.7414432148 + 0.1547169611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.163 + 0.986i)T \)
3 \( 1 + (-0.999 + 0.0299i)T \)
5 \( 1 + (0.971 - 0.237i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (-0.873 - 0.486i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (-0.393 + 0.919i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.599 - 0.800i)T \)
41 \( 1 + (0.473 - 0.880i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.712 - 0.701i)T \)
53 \( 1 + (0.0149 - 0.999i)T \)
59 \( 1 + (0.525 - 0.850i)T \)
61 \( 1 + (0.0149 + 0.999i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (0.712 + 0.701i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.936 + 0.351i)T \)
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

−23.22571217564468057858090669180, −22.39831198146469858122071156424, −21.78726505167667782953470431262, −21.12145168380456296629419077915, −20.32927927054145647710793558127, −19.03782453498315413111409783206, −18.399347730597998512836055288229, −17.72677972343301172666271140957, −17.04270969616316403232928797025, −16.17468419251194737656336748311, −14.83514498896195982307409706114, −13.60702849720589567282842158294, −13.18631790982972195067388477247, −12.1445477396177018236376060781, −11.28141205311944147265916337211, −10.58815694411550359271866295865, −9.846406344011125187164881491481, −8.97774569650946037501181440554, −7.7289324094751892825452491724, −6.30961494135660493269601094958, −5.725099563768378831939738583324, −4.528118531109999397378168804735, −3.58622834060117514617487195245, −2.04168410680163144105294033882, −1.36470773500424247103455688127, 0.5526839846432739689060945419, 1.90336075270141713751668349729, 3.88086171171546049275202516785, 5.019541809297920977036501629531, 5.586333555715855640570809295716, 6.52600442018082071958515938743, 7.127816651202996219331325991758, 8.61237327606754399818583663026, 9.27641239365325470715230414190, 10.361357796137125591688036590255, 11.00719496538181372454466722392, 12.47976795747934332601410549392, 13.22334593257961372187480820714, 13.914359068928336806461533594302, 15.09946424412999965684814875892, 16.03891750127314105676604645684, 16.52960544097874512341201845148, 17.58821550834706229810828616565, 17.95022171849828059418911722351, 18.62406280798321422312835877824, 20.02847424988232089326313255664, 21.132635228335821766542081929587, 22.151850063210602095537389595978, 22.420808748612621851329379653361, 23.60243372050289237758751549447

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