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Properties

Label 1-539-539.93-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.738 + 0.674i$
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  + (−0.599 + 0.800i)2-s + (−0.772 + 0.635i)3-s + (−0.280 − 0.959i)4-s + (0.712 + 0.701i)5-s + (−0.0448 − 0.998i)6-s + (0.936 + 0.351i)8-s + (0.193 − 0.981i)9-s + (−0.988 + 0.149i)10-s + (0.826 + 0.563i)12-s + (−0.393 − 0.919i)13-s + (−0.995 − 0.0896i)15-s + (−0.842 + 0.538i)16-s + (−0.646 + 0.762i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (0.473 − 0.880i)20-s + ⋯
L(s)  = 1  + (−0.599 + 0.800i)2-s + (−0.772 + 0.635i)3-s + (−0.280 − 0.959i)4-s + (0.712 + 0.701i)5-s + (−0.0448 − 0.998i)6-s + (0.936 + 0.351i)8-s + (0.193 − 0.981i)9-s + (−0.988 + 0.149i)10-s + (0.826 + 0.563i)12-s + (−0.393 − 0.919i)13-s + (−0.995 − 0.0896i)15-s + (−0.842 + 0.538i)16-s + (−0.646 + 0.762i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (0.473 − 0.880i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.738 + 0.674i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.738 + 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2713698269 + 0.6993118485i\)
\(L(\frac12)\) \(\approx\) \(0.2713698269 + 0.6993118485i\)
\(L(1)\) \(\approx\) \(0.5372603040 + 0.4232169928i\)
\(L(1)\) \(\approx\) \(0.5372603040 + 0.4232169928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.599 + 0.800i)T \)
3 \( 1 + (-0.772 + 0.635i)T \)
5 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (-0.393 - 0.919i)T \)
17 \( 1 + (-0.646 + 0.762i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.791 - 0.611i)T \)
41 \( 1 + (0.936 + 0.351i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.575 + 0.817i)T \)
53 \( 1 + (-0.337 + 0.941i)T \)
59 \( 1 + (-0.163 + 0.986i)T \)
61 \( 1 + (-0.337 - 0.941i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (0.575 - 0.817i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−22.86140084530041907525954617920, −22.192280188480111116448163350904, −21.359449457259531557651223446947, −20.579079698215181432600305518577, −19.69861869731904751738080231016, −18.75369314330494534597577091234, −18.1071245465520940967106487425, −17.349401416941651280863040453923, −16.62288681899146811161017627114, −16.052975158207460712554608260947, −14.06737251790755247269544123590, −13.50010959772617303556463573873, −12.474162304405560274062711069702, −12.00372364458687393306629956118, −11.06799743027804664451570549366, −10.08696828637490778653507186180, −9.290465372132810476054423948461, −8.3394022354676013430436216627, −7.27975927486253363739320193379, −6.33711538393285403002099722821, −5.09176279078091803653262446884, −4.292771741977164798640382314392, −2.53001768327802838342375484189, −1.749648525315696466014135731298, −0.62984166476023917469107606834, 1.15596881467780307085460234256, 2.74692608788323299647803082743, 4.2465916563636885229872481484, 5.413077358370410997146710970537, 5.93028296154513195562666294760, 6.885530471038210802141115517896, 7.778406047593960589655268389289, 9.28160326912891417148862596441, 9.64967609196788745144513390135, 10.812156294637692746052388764249, 11.06092427526604535925805120035, 12.69075374721168702513084746926, 13.710052385478428821843578184724, 14.80298181639424518385192898335, 15.298427955176983161920490958796, 16.16768140261775577720182414128, 17.18580863747272931793534551074, 17.72206333389343531689770370360, 18.20283564854543230490127813160, 19.415419755094575173944070338276, 20.30130201832321012960261590689, 21.57368927713674916414409150944, 22.19750970202190688069039870975, 22.86619524038233627412200055666, 23.78621974746616414245323611110

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