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Properties

Label 1-539-539.191-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.963 - 0.268i$
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  + (0.575 + 0.817i)2-s + (0.251 − 0.967i)3-s + (−0.337 + 0.941i)4-s + (−0.447 + 0.894i)5-s + (0.936 − 0.351i)6-s + (−0.963 + 0.266i)8-s + (−0.873 − 0.486i)9-s + (−0.988 + 0.149i)10-s + (0.826 + 0.563i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (−0.772 − 0.635i)16-s + (−0.925 − 0.379i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + (−0.691 − 0.722i)20-s + ⋯
L(s)  = 1  + (0.575 + 0.817i)2-s + (0.251 − 0.967i)3-s + (−0.337 + 0.941i)4-s + (−0.447 + 0.894i)5-s + (0.936 − 0.351i)6-s + (−0.963 + 0.266i)8-s + (−0.873 − 0.486i)9-s + (−0.988 + 0.149i)10-s + (0.826 + 0.563i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (−0.772 − 0.635i)16-s + (−0.925 − 0.379i)17-s + (−0.104 − 0.994i)18-s + (−0.104 + 0.994i)19-s + (−0.691 − 0.722i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.963 - 0.268i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.963 - 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06652600009 + 0.4867184483i\)
\(L(\frac12)\) \(\approx\) \(-0.06652600009 + 0.4867184483i\)
\(L(1)\) \(\approx\) \(0.8317466055 + 0.4083924344i\)
\(L(1)\) \(\approx\) \(0.8317466055 + 0.4083924344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.575 + 0.817i)T \)
3 \( 1 + (0.251 - 0.967i)T \)
5 \( 1 + (-0.447 + 0.894i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (-0.925 - 0.379i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.999 + 0.0299i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.946 - 0.323i)T \)
53 \( 1 + (0.791 + 0.611i)T \)
59 \( 1 + (0.712 - 0.701i)T \)
61 \( 1 + (0.791 - 0.611i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (-0.946 + 0.323i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−22.567712191192155941109091496835, −22.115405656471653862420764513421, −21.23217093777394155601794169584, −20.513986653360236586663884694386, −19.74535820914327043089699953552, −19.429231248085645178884104245976, −17.9164241096487897800371344871, −16.94502367203578624551260205362, −15.973360623201802630771551995350, −15.20402108141104241263428244038, −14.508023118177577527459897670423, −13.44362202040815679660118065071, −12.61314945451513104410400634716, −11.765058123574096282816084795460, −10.84996382432521941648044594644, −10.095092873723987143074889040981, −8.98474470791052995138814508608, −8.62171302575263204206509063283, −6.96115383386610743984171588801, −5.4411976146611393492601126382, −4.7884389850149336643772207794, −4.119401772135119208619077295099, −3.04387537959908857438707127410, −1.9691687141961342298215763630, −0.188006521621204196017225233955, 2.11474341590695189274112775803, 3.06447115987423479159155497089, 4.038758304049029335485984052161, 5.38216862605506763919864776909, 6.40221868238287198794682406280, 7.132743085381892500083851860329, 7.72538597041006400960689739430, 8.638230262298707050562212303755, 9.850219096832686980736870804945, 11.40260224589426308320571740921, 11.9579561609498078492273513960, 12.93572125675234335534742702968, 13.77927666255989321692695978212, 14.53339950455270299203756970182, 15.12585404995669016781787893647, 16.11842545071026913916899223924, 17.2876107097533530561726183408, 17.86149273202763799620304411373, 18.7716102984344908834959958458, 19.51688365817256039483295513627, 20.5118662616939813022298962008, 21.7128062715486894610910511717, 22.485748578556676232839011534682, 23.16913923254976336662710530982, 23.85391664167756145343138401095

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