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Properties

Label 1-539-539.124-r1-0-0
Degree $1$
Conductor $539$
Sign $-0.999 + 0.0168i$
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  + (−0.447 + 0.894i)2-s + (0.337 − 0.941i)3-s + (−0.599 − 0.800i)4-s + (0.925 − 0.379i)5-s + (0.691 + 0.722i)6-s + (0.983 − 0.178i)8-s + (−0.772 − 0.635i)9-s + (−0.0747 + 0.997i)10-s + (−0.955 + 0.294i)12-s + (0.550 + 0.834i)13-s + (−0.0448 − 0.998i)15-s + (−0.280 + 0.959i)16-s + (−0.420 + 0.907i)17-s + (0.913 − 0.406i)18-s + (−0.913 − 0.406i)19-s + (−0.858 − 0.512i)20-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)2-s + (0.337 − 0.941i)3-s + (−0.599 − 0.800i)4-s + (0.925 − 0.379i)5-s + (0.691 + 0.722i)6-s + (0.983 − 0.178i)8-s + (−0.772 − 0.635i)9-s + (−0.0747 + 0.997i)10-s + (−0.955 + 0.294i)12-s + (0.550 + 0.834i)13-s + (−0.0448 − 0.998i)15-s + (−0.280 + 0.959i)16-s + (−0.420 + 0.907i)17-s + (0.913 − 0.406i)18-s + (−0.913 − 0.406i)19-s + (−0.858 − 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.999 + 0.0168i$
Analytic conductor: \(57.9235\)
Root analytic conductor: \(57.9235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (1:\ ),\ -0.999 + 0.0168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001310816089 - 0.1557519799i\)
\(L(\frac12)\) \(\approx\) \(0.001310816089 - 0.1557519799i\)
\(L(1)\) \(\approx\) \(0.8292924249 + 0.01027183556i\)
\(L(1)\) \(\approx\) \(0.8292924249 + 0.01027183556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.447 + 0.894i)T \)
3 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (0.925 - 0.379i)T \)
13 \( 1 + (0.550 + 0.834i)T \)
17 \( 1 + (-0.420 + 0.907i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.946 - 0.323i)T \)
41 \( 1 + (-0.983 + 0.178i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.887 + 0.460i)T \)
53 \( 1 + (0.575 - 0.817i)T \)
59 \( 1 + (0.646 - 0.762i)T \)
61 \( 1 + (-0.575 - 0.817i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.887 - 0.460i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.550 - 0.834i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−23.16585861616434029450423448336, −22.424830016979760579864890634462, −21.7902601797902940271270873044, −21.05336448025223629856983621323, −20.3946028116091137916232076173, −19.70335181495054836403355139141, −18.487337110910958571120526298135, −17.93900647650767660781789047551, −16.965019029722054627314973550477, −16.20894953710630099512627980526, −15.078703352896952330687664843963, −14.054632081059274768960021746281, −13.477920757777104802003834264813, −12.38898479059966156320155402594, −11.21819180097311970769445961624, −10.30675970524767260119352439168, −10.130042831670928099284987615973, −8.84761070350491548184188980559, −8.43980706266476096640207976492, −6.954854071650717059893495239819, −5.57762137045168143286683366616, −4.63211610780300983918769385190, −3.45607570798055398506659283971, −2.70868259927946412726749682508, −1.6706760775574064246725526994, 0.03988419722939132503166095136, 1.47309726280928414484814528752, 2.08254563144172204372259932231, 3.93587495623008254052588608637, 5.22567334714045659999248100246, 6.356927132186227659407793469118, 6.56630909878548727967320355511, 7.95301238297725094002704109045, 8.6465374939613143246671293207, 9.35625716615623096974581493363, 10.387332398807596240427590899673, 11.61632676564255250854373055253, 12.97436238216833051803532446272, 13.42904950168635064085665161340, 14.236486722570356680859299465549, 15.04181044132626084650241886671, 16.17200066529657151141826037739, 17.16047527680458202781980084739, 17.60504826351055877510158434330, 18.45110218349355304323038826155, 19.24764137716047855899732148631, 19.98117313998627000390277642331, 21.1351808264986190105585728760, 22.032024775165377461587687415549, 23.2592064001266712252529125834

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