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Properties

Label 1-539-539.323-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.857 + 0.514i$
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  + (−0.550 + 0.834i)2-s + (0.983 + 0.178i)3-s + (−0.393 − 0.919i)4-s + (0.134 + 0.990i)5-s + (−0.691 + 0.722i)6-s + (0.983 + 0.178i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.691 + 0.722i)16-s + (−0.995 − 0.0896i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.858 − 0.512i)20-s + ⋯
L(s)  = 1  + (−0.550 + 0.834i)2-s + (0.983 + 0.178i)3-s + (−0.393 − 0.919i)4-s + (0.134 + 0.990i)5-s + (−0.691 + 0.722i)6-s + (0.983 + 0.178i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.691 + 0.722i)16-s + (−0.995 − 0.0896i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.858 − 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3280944116 + 1.184499666i\)
\(L(\frac12)\) \(\approx\) \(0.3280944116 + 1.184499666i\)
\(L(1)\) \(\approx\) \(0.7855654546 + 0.6726140657i\)
\(L(1)\) \(\approx\) \(0.7855654546 + 0.6726140657i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.550 + 0.834i)T \)
3 \( 1 + (0.983 + 0.178i)T \)
5 \( 1 + (0.134 + 0.990i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (-0.995 - 0.0896i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.753 + 0.657i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.0448 - 0.998i)T \)
53 \( 1 + (-0.995 + 0.0896i)T \)
59 \( 1 + (0.983 - 0.178i)T \)
61 \( 1 + (-0.995 - 0.0896i)T \)
67 \( 1 + T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (-0.0448 + 0.998i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.550 - 0.834i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−22.93774090098652481070207018547, −21.89318755545031158552780912176, −21.04066375910185408609235141427, −20.48319806583340400502405265327, −19.76764571157871131576759379922, −19.20182088569305841039081147966, −18.108875880104254312656534231955, −17.40284903373700333309828181839, −16.45057552181318652684126341455, −15.49227395497364261793511993626, −14.38495563514768891201580171414, −13.309091442721474155151062053174, −12.79084324494585904013095608253, −12.11299449329347772594762264823, −10.78165108844626403508992993916, −9.83346405262527353939908075546, −9.16651367820234176463344185309, −8.21500212937140864364873055179, −7.84611035468406342038307533412, −6.3548532347126212239458404452, −4.66445788950535756646109684251, −4.041524838978908859135895480976, −2.678550629171399356571863805179, −1.977285927520217048241790227821, −0.677799339101842890867350956609, 1.74042941492036415134772086058, 2.67487706993569450746454494046, 4.03755546473269304684081317273, 5.005037857087278748310113950300, 6.58477219364254420159751363355, 6.94316431204829169712636158474, 7.9996100430374514130195555698, 8.90068954995424030204454858810, 9.672868615254320077387158952499, 10.463046322859062372394269716608, 11.40749197679164694585830497914, 13.07267313985432975085253484627, 13.94494580084930785419605854011, 14.47836321696505645007569942303, 15.32969221947711341573329462384, 15.87043662342273241806889406731, 17.08137764893592024624378951612, 17.94560029655905083214635156019, 18.71157665438485444903953078885, 19.530201307805605474506035320492, 19.95842102732636156499506392212, 21.56465103315912150974064816264, 21.89331921013266802385157470333, 23.18050678915305623231198699591, 23.93442071022954710586484377709

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