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Properties

Label 1-539-539.137-r0-0-0
Degree $1$
Conductor $539$
Sign $0.796 + 0.604i$
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  + (−0.946 + 0.323i)2-s + (0.998 + 0.0598i)3-s + (0.791 − 0.611i)4-s + (0.887 + 0.460i)5-s + (−0.963 + 0.266i)6-s + (−0.550 + 0.834i)8-s + (0.992 + 0.119i)9-s + (−0.988 − 0.149i)10-s + (0.826 − 0.563i)12-s + (0.753 + 0.657i)13-s + (0.858 + 0.512i)15-s + (0.251 − 0.967i)16-s + (0.525 − 0.850i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + (0.983 − 0.178i)20-s + ⋯
L(s)  = 1  + (−0.946 + 0.323i)2-s + (0.998 + 0.0598i)3-s + (0.791 − 0.611i)4-s + (0.887 + 0.460i)5-s + (−0.963 + 0.266i)6-s + (−0.550 + 0.834i)8-s + (0.992 + 0.119i)9-s + (−0.988 − 0.149i)10-s + (0.826 − 0.563i)12-s + (0.753 + 0.657i)13-s + (0.858 + 0.512i)15-s + (0.251 − 0.967i)16-s + (0.525 − 0.850i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + (0.983 − 0.178i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.521355385 + 0.5116096623i\)
\(L(\frac12)\) \(\approx\) \(1.521355385 + 0.5116096623i\)
\(L(1)\) \(\approx\) \(1.175314653 + 0.2586166081i\)
\(L(1)\) \(\approx\) \(1.175314653 + 0.2586166081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.946 + 0.323i)T \)
3 \( 1 + (0.998 + 0.0598i)T \)
5 \( 1 + (0.887 + 0.460i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (0.525 - 0.850i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.280 - 0.959i)T \)
41 \( 1 + (-0.550 + 0.834i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.0149 + 0.999i)T \)
53 \( 1 + (-0.999 + 0.0299i)T \)
59 \( 1 + (-0.447 + 0.894i)T \)
61 \( 1 + (-0.999 - 0.0299i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.983 + 0.178i)T \)
73 \( 1 + (0.0149 - 0.999i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.753 - 0.657i)T \)
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

−23.62917483199147061865980077815, −22.104004427775512667271639965431, −21.08464670445721477251137874338, −20.92158478443487018973405775518, −19.965313694921370751809017682171, −19.12533369359170474691729259561, −18.46389342462194505707034242692, −17.44621097074029208882430859058, −16.87592852833884508874418000354, −15.67557900901392286480660198882, −15.040510686117871002760518886340, −13.75628067586972470108424089381, −13.04132346114592968667251357030, −12.30627043274355218621863867464, −10.90630557338542940003026004624, −10.03545191218017662357885065529, −9.443071380323360951489342118059, −8.39585813739395504864549590225, −8.03558828926227001094704906579, −6.69431437286355944310642544075, −5.753929933906614414241445295974, −4.0836283859382123136133299967, −3.08672173848216187341352094173, −1.99192474016364454129377915266, −1.23993249287714145236201746270, 1.380760947760237968616285788329, 2.28840022526795417666155423096, 3.19381461761166608177224506255, 4.751682417504068613583437389192, 6.14275668101814157781959366403, 6.847300899117023653253175653199, 7.79398913568473336626475575837, 8.9063712127965822212204610380, 9.285051487463248936697667161395, 10.35365578241205881304357832332, 10.941134706910317503510924133211, 12.37803037957918862483524654978, 13.60393402051334432477561825359, 14.284165270991999698821187062553, 14.960952389125395551730528857447, 15.97783158675086166369991053201, 16.73442468145372375090562663487, 17.79561969885915469425407704097, 18.62558284915072626393933506994, 19.001336856513833553316700889418, 20.11511863461462169060560319382, 20.94464175757938036005106361467, 21.387827217572718648811014564753, 22.722863199499368050659557508405, 23.84741992993185987345395161126

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