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Properties

Label 1-539-539.10-r0-0-0
Degree $1$
Conductor $539$
Sign $0.768 + 0.640i$
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  + (−0.955 − 0.294i)2-s + (−0.365 + 0.930i)3-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (−0.365 + 0.930i)3-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.768 + 0.640i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.768 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8438972292 + 0.3054525351i\)
\(L(\frac12)\) \(\approx\) \(0.8438972292 + 0.3054525351i\)
\(L(1)\) \(\approx\) \(0.7435506577 + 0.1290943301i\)
\(L(1)\) \(\approx\) \(0.7435506577 + 0.1290943301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T \)
3 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 - T \)
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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

−23.69066520787980150772051312991, −22.62655722208028413717863997703, −21.576737371288330727981889793821, −20.71515647669167735304473567180, −19.58602662883743988843618547749, −18.98892930700831845614627401643, −18.146736565441377155270479756622, −17.527619859684580373723453942734, −16.911711222212793384322544607017, −16.0353780408906519387299766959, −14.79703546058058372030947421747, −13.88296450371919331000296120237, −13.20812808215717072897052045919, −11.81061840697097912625314939905, −11.32910429110525308423840940283, −10.149720469088688069246034622057, −9.35083964431599259727717577234, −8.46459130896879160108893180688, −7.277981281127528897716687236893, −6.713806992596250310638357555623, −5.88239479011938974873783416025, −4.92079621561193173706110119096, −2.67430294198357007056381161106, −2.01454626168758718200206869551, −0.84902088566310917550966422420, 1.03692764242317159014919542986, 2.40602573752610414296589735671, 3.40237101350971360216411261904, 4.71028170066804136274615746338, 5.90635106268328958042027762719, 6.52648831900963526026504175064, 8.14426144659661750623458681917, 8.768668276675689184706430158115, 9.915785274427967383475373746473, 10.27329237440149418057552422539, 11.0022667446454677175493322063, 12.28203186045137633394021661293, 12.894926147029649206866046006515, 14.45755569373584897399965086274, 15.1240767134134099043883165701, 16.338555854373549101883283597759, 16.73639590726488376714836110345, 17.72909195451741863481060045029, 18.0753482321621345761335283774, 19.45285230213427729671228519947, 20.22347280036229893742540537301, 21.11655797130119869135194797569, 21.50735701788872973178440958999, 22.39637316908656119788627608490, 23.41199924994369854012400331041

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