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Properties

Label 1-539-539.36-r0-0-0
Degree $1$
Conductor $539$
Sign $0.852 - 0.522i$
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  + (0.134 + 0.990i)2-s + (−0.393 − 0.919i)3-s + (−0.963 + 0.266i)4-s + (−0.995 + 0.0896i)5-s + (0.858 − 0.512i)6-s + (−0.393 − 0.919i)8-s + (−0.691 + 0.722i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)12-s + (0.134 + 0.990i)13-s + (0.473 + 0.880i)15-s + (0.858 − 0.512i)16-s + (−0.550 + 0.834i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.936 − 0.351i)20-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (−0.393 − 0.919i)3-s + (−0.963 + 0.266i)4-s + (−0.995 + 0.0896i)5-s + (0.858 − 0.512i)6-s + (−0.393 − 0.919i)8-s + (−0.691 + 0.722i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)12-s + (0.134 + 0.990i)13-s + (0.473 + 0.880i)15-s + (0.858 − 0.512i)16-s + (−0.550 + 0.834i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.936 − 0.351i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5707111401 - 0.1608743622i\)
\(L(\frac12)\) \(\approx\) \(0.5707111401 - 0.1608743622i\)
\(L(1)\) \(\approx\) \(0.6592981393 + 0.1204734736i\)
\(L(1)\) \(\approx\) \(0.6592981393 + 0.1204734736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.134 + 0.990i)T \)
3 \( 1 + (-0.393 - 0.919i)T \)
5 \( 1 + (-0.995 + 0.0896i)T \)
13 \( 1 + (0.134 + 0.990i)T \)
17 \( 1 + (-0.550 + 0.834i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.0448 - 0.998i)T \)
41 \( 1 + (-0.393 - 0.919i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.473 - 0.880i)T \)
53 \( 1 + (-0.550 - 0.834i)T \)
59 \( 1 + (-0.393 + 0.919i)T \)
61 \( 1 + (-0.550 + 0.834i)T \)
67 \( 1 + T \)
71 \( 1 + (0.936 + 0.351i)T \)
73 \( 1 + (0.473 + 0.880i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.134 - 0.990i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
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

−23.247816937180489448585201354872, −22.55321650355674294804473808837, −21.82652644408896000071598274864, −20.96901266949235447466459175653, −20.09714252051277655783106617694, −19.73002394837044945213853378030, −18.53869724612404923968920752833, −17.68656221669356197248550214610, −16.8138830760289428084365103499, −15.614775958151332710397616153934, −15.19687390070897894403565501134, −14.109531168460296392698562869528, −12.91775287068481453214039971420, −12.17959473348963900022319858366, −11.07167047532896737152599266508, −10.95988925778592886332447620860, −9.72413570057245569219277876505, −8.898676004941408761554569913067, −8.028064221421906354950383270493, −6.52211422599893945624346678670, −5.04562916955608824046250483737, −4.698368921652462070585991426995, −3.46657530413988145159407617137, −2.90611775218965597664909202581, −0.96336385675873517021718399022, 0.41654479396899573287860411870, 2.13046970445572741504993516177, 3.79285323685632670202724164035, 4.54999424762421109629028777662, 5.83903441682742633429456445498, 6.64485462021222412033401403212, 7.32382693420669513271089275325, 8.28806657663919527183659448953, 8.83805835076351191265429555894, 10.45938727578810018788814907990, 11.50869864968367488746962914772, 12.35902719117933183042406512893, 13.07739642241547671947979607597, 14.0447138123891096242450965677, 14.89719349263809807108154134690, 15.71573384843219571726301707644, 16.819591237732229104800077251086, 17.1057466705457098585892980930, 18.4126718921088589441433362039, 18.92396932634511801463707759881, 19.57077679563009891424878496892, 20.96556696956714918915956980355, 22.10680505088229394472351343516, 22.897723135571585684504114656979, 23.44958429063665239058396355590

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