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Properties

Label 1-539-539.248-r0-0-0
Degree $1$
Conductor $539$
Sign $0.164 - 0.986i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1823802502 - 0.1544952940i\)
\(L(\frac12)\) \(\approx\) \(0.1823802502 - 0.1544952940i\)
\(L(1)\) \(\approx\) \(0.5105907834 + 0.1367553486i\)
\(L(1)\) \(\approx\) \(0.5105907834 + 0.1367553486i\)

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

−23.83379378469466776970748780525, −23.347793568953439992061651836097, −21.64955295408805818686211719238, −20.83713527543585926748085385398, −19.8582873648919664540921430749, −19.40930659017956599931682431205, −18.70203591940280873969283687032, −17.62291192772924420339476634760, −17.02075174652091905559669108249, −16.36132189045523385227811501416, −15.18529256916879687592680022319, −14.37155512921632071425992261497, −13.06284778315206751306226992802, −12.42431103584179730207996158874, −11.57871002083368529482938676369, −10.67009216858642439706709470752, −9.22838802298065248086139307683, −8.77950980865184191085978957719, −7.92000835086883519117481149624, −7.03325300344991421058824689672, −6.20486154947227687684574957578, −5.04702210391928711166091464160, −3.49765627007147573654495976788, −1.99742188345216051466596931219, −1.40259217112663248779016031180, 0.16476979519098340892948149776, 2.36304557693558325736042761744, 2.97539119489240563873410693859, 4.083112820346783284164451319347, 5.47261764569546422025916502389, 6.61692931445104686292910933980, 7.55825503759830290154993885687, 8.469681670505900272127095200629, 9.42708390514017715894454686388, 10.21348295416212388179801205678, 10.933451887175641118510841999531, 11.498384221757510853916689316823, 12.777553411710332470937447047637, 14.21772659671187049914122185722, 15.12411390226685439455956014076, 15.503777702509812200440782409345, 16.53226084222382774735421429481, 17.368343286645162953780821241651, 18.15296595345038074556068736659, 19.14781435038532005473137639178, 19.77130766133151314685030751063, 20.67999955345566033053728268599, 21.35969253231671834304238160126, 22.39795686901493221513453041086, 22.9015572146966858499113928772

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