Properties

Label 1-539-539.145-r0-0-0
Degree $1$
Conductor $539$
Sign $0.996 + 0.0805i$
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  + (0.337 + 0.941i)2-s + (0.873 − 0.486i)3-s + (−0.772 + 0.635i)4-s + (0.599 − 0.800i)5-s + (0.753 + 0.657i)6-s + (−0.858 − 0.512i)8-s + (0.525 − 0.850i)9-s + (0.955 + 0.294i)10-s + (−0.365 + 0.930i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.193 − 0.981i)16-s + (0.712 − 0.701i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.0448 + 0.998i)20-s + ⋯
L(s)  = 1  + (0.337 + 0.941i)2-s + (0.873 − 0.486i)3-s + (−0.772 + 0.635i)4-s + (0.599 − 0.800i)5-s + (0.753 + 0.657i)6-s + (−0.858 − 0.512i)8-s + (0.525 − 0.850i)9-s + (0.955 + 0.294i)10-s + (−0.365 + 0.930i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.193 − 0.981i)16-s + (0.712 − 0.701i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.0448 + 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.996 + 0.0805i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.996 + 0.0805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.259759238 + 0.09111414165i\)
\(L(\frac12)\) \(\approx\) \(2.259759238 + 0.09111414165i\)
\(L(1)\) \(\approx\) \(1.661683583 + 0.2399924919i\)
\(L(1)\) \(\approx\) \(1.661683583 + 0.2399924919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.337 + 0.941i)T \)
3 \( 1 + (0.873 - 0.486i)T \)
5 \( 1 + (0.599 - 0.800i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (0.712 - 0.701i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.550 - 0.834i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.998 + 0.0598i)T \)
41 \( 1 + (0.858 + 0.512i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.791 + 0.611i)T \)
53 \( 1 + (0.251 - 0.967i)T \)
59 \( 1 + (-0.0149 - 0.999i)T \)
61 \( 1 + (0.251 + 0.967i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (0.791 + 0.611i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
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

−23.21535924048395750929871255969, −22.234350921223441423049460100400, −21.553565061328308764569296754845, −21.08354729329838156580061632576, −20.13324792531283548647663191378, −19.383548397441431496878593039094, −18.56568214251537642327517653726, −17.92846952065159709721190409082, −16.63587532430154382875428059504, −15.35151933367588931490042295933, −14.708372778946078252529358589526, −13.96306773291303710672120028775, −13.29939893967020116179187426142, −12.37655036240105719343129572669, −10.8922623883983970824724642935, −10.599694918014071099693200617332, −9.65641747826637106320733310974, −8.83911336306346732649832517671, −7.85589249217649848243950320191, −6.34515759602424567502352651024, −5.44635050488101659143791396636, −4.082489503030652796650187222923, −3.45518864582734436493680170191, −2.4345848381405732437556867922, −1.603765594059114711890775751061, 1.07825233612375468984909725668, 2.43948464405647394729736838485, 3.72931959423562883846360850245, 4.59302335210103122826046466680, 5.88895281051612621562979981120, 6.486097666019709183337000709903, 7.78089507921057857945070584040, 8.35774952136560666895942340501, 9.200678805758377085729674955860, 9.948288325989683777924272195577, 11.75757183887655608179494112832, 12.80006376133161329625540937041, 13.21876555091626905639173663612, 14.158570898564883160943652374257, 14.65880066574302085681375239837, 15.96747047478524692880602806259, 16.38088072042259088521136785148, 17.6162003872284521802266181562, 18.1755084715316816732038946057, 19.10890303208696782371476346029, 20.2498344959244123105179572534, 21.08495559121981651054065479721, 21.54511267392869882713954173855, 22.9439001546172299074238511121, 23.65717694493651801920456954863

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