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Properties

Label 1-539-539.428-r1-0-0
Degree $1$
Conductor $539$
Sign $0.404 - 0.914i$
Analytic cond. $57.9235$
Root an. cond. $57.9235$
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  + (0.900 − 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 + 0.974i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s − 18-s − 19-s + (0.623 + 0.781i)20-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + (0.222 + 0.974i)6-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (0.623 + 0.781i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s − 18-s − 19-s + (0.623 + 0.781i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(57.9235\)
Root analytic conductor: \(57.9235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (1:\ ),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.031392609 - 1.322290055i\)
\(L(\frac12)\) \(\approx\) \(2.031392609 - 1.322290055i\)
\(L(1)\) \(\approx\) \(1.504882255 - 0.1085117663i\)
\(L(1)\) \(\approx\) \(1.504882255 - 0.1085117663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + T \)
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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

−23.42453014337319750681319094104, −22.984304795636716606332565474608, −21.68292838957168493399778643275, −21.05182019946400626733362437333, −19.99459090128353050868593792427, −19.40604536167955064978017199683, −18.17647200106325328198186129569, −17.16513469892305455328885616416, −16.70716141848674347679055489074, −15.7090029179052348321707132193, −14.81321903449717386050711230522, −13.652459939171889039097363695326, −13.11716321323044936984993651074, −12.50228184710410605653205432030, −11.571868507482218672260441426746, −10.879007851644446334039225220235, −8.90459289950448592039936779175, −8.33917932591726294297065710824, −7.35848875630968180333549387667, −6.38516418675723024708743268685, −5.67379740389302032659430761340, −4.61535880470300407011804215792, −3.66392718580480338690117438847, −2.21498337091042129889919488828, −1.22395748947700057573990089465, 0.4585243069539443802849660066, 2.35951567245441083712679091027, 3.188000376360003571743713510104, 4.07903468322066344847596999480, 4.904273780956944462681348735798, 6.12903710449821014773260882531, 6.64672080845995124905147946447, 8.15864461072052208941396203726, 9.47486879064568904947359443060, 10.358782787020342215245741831674, 11.09340567997508410430722832779, 11.50932479103400344960595985512, 12.78629694122758675289774637371, 13.791958039932924823019227654078, 14.606876397627318608672741773877, 15.36658116827267741619421355189, 15.82987007345722715326334965941, 16.98088361662961254478171879131, 18.16074921192740494356923661148, 19.02880705787754670291677992930, 19.968441737198197239627009736769, 20.85176744636700699331662530816, 21.369221774630813242779485561649, 22.45845977852161381454567156905, 22.74804107822343153446162630758

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