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Properties

Label 1-539-539.150-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.526 - 0.850i$
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  + (−0.420 + 0.907i)2-s + (−0.712 − 0.701i)3-s + (−0.646 − 0.762i)4-s + (−0.998 + 0.0598i)5-s + (0.936 − 0.351i)6-s + (0.963 − 0.266i)8-s + (0.0149 + 0.999i)9-s + (0.365 − 0.930i)10-s + (−0.0747 + 0.997i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (−0.163 + 0.986i)16-s + (0.791 − 0.611i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (0.691 + 0.722i)20-s + ⋯
L(s)  = 1  + (−0.420 + 0.907i)2-s + (−0.712 − 0.701i)3-s + (−0.646 − 0.762i)4-s + (−0.998 + 0.0598i)5-s + (0.936 − 0.351i)6-s + (0.963 − 0.266i)8-s + (0.0149 + 0.999i)9-s + (0.365 − 0.930i)10-s + (−0.0747 + 0.997i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (−0.163 + 0.986i)16-s + (0.791 − 0.611i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (0.691 + 0.722i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09263798052 - 0.1663689117i\)
\(L(\frac12)\) \(\approx\) \(0.09263798052 - 0.1663689117i\)
\(L(1)\) \(\approx\) \(0.4607401854 + 0.04166142793i\)
\(L(1)\) \(\approx\) \(0.4607401854 + 0.04166142793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.420 + 0.907i)T \)
3 \( 1 + (-0.712 - 0.701i)T \)
5 \( 1 + (-0.998 + 0.0598i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (0.791 - 0.611i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (-0.473 + 0.880i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.525 + 0.850i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.193 - 0.981i)T \)
53 \( 1 + (-0.925 + 0.379i)T \)
59 \( 1 + (-0.251 - 0.967i)T \)
61 \( 1 + (-0.925 - 0.379i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.193 - 0.981i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−23.35662816947668812207171949623, −22.697872419562468653676452286067, −22.024158855646122714548030813123, −21.12322256721667622436900908405, −20.41197733460636428285713691100, −19.542465974487277567363286011031, −18.81138941937602159776772003150, −17.84357827296725694152173803868, −16.893143830992269056617837520443, −16.43269230585028688707364475757, −15.284445753066340349929273281938, −14.50821232028231959418012753769, −13.00621937416220710589707463350, −12.18631763975477474121004554099, −11.58985108669594178408078359150, −10.854429832783725460757105700336, −9.88225050910508443988476334747, −9.24981870527191179963649041125, −7.974756642239744707187999975817, −7.271674760808005177811500928283, −5.61797774860972605207830265155, −4.65847567818033235042639436434, −3.7926570884236521456495230769, −2.98309385228641994267237887647, −1.23680815855538387049365407510, 0.1542631251700477897766383203, 1.40138846892428393309419143715, 3.158739446971052031530035244000, 4.838508388565087664215755846591, 5.19877853981796239799431923565, 6.64248709728187915492731975757, 7.24465035909062624893302223047, 7.856152845443539516533857320975, 8.93067114586371856415057774029, 10.086206064584616279475860344541, 11.07324930296727743265196525076, 11.96331696414374730021729377606, 12.78996804197606157819699329397, 13.904390369321715719278582699990, 14.76258261479115356026639617064, 15.66938079826501046755534002530, 16.625072606623192420222434405015, 16.95475929300282466962138248949, 18.297664249731904063393610298415, 18.56102738299264972459177134965, 19.54871690457664975460922954760, 20.213820996852200654959009812971, 22.01406217463194607636997762363, 22.5368457249329000318775153615, 23.49708790425949434026989896517

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