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Properties

Label 1-539-539.4-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.144 - 0.989i$
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  + (−0.772 + 0.635i)2-s + (0.525 − 0.850i)3-s + (0.193 − 0.981i)4-s + (−0.280 − 0.959i)5-s + (0.134 + 0.990i)6-s + (0.473 + 0.880i)8-s + (−0.447 − 0.894i)9-s + (0.826 + 0.563i)10-s + (−0.733 − 0.680i)12-s + (0.936 + 0.351i)13-s + (−0.963 − 0.266i)15-s + (−0.925 − 0.379i)16-s + (0.0149 − 0.999i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.995 + 0.0896i)20-s + ⋯
L(s)  = 1  + (−0.772 + 0.635i)2-s + (0.525 − 0.850i)3-s + (0.193 − 0.981i)4-s + (−0.280 − 0.959i)5-s + (0.134 + 0.990i)6-s + (0.473 + 0.880i)8-s + (−0.447 − 0.894i)9-s + (0.826 + 0.563i)10-s + (−0.733 − 0.680i)12-s + (0.936 + 0.351i)13-s + (−0.963 − 0.266i)15-s + (−0.925 − 0.379i)16-s + (0.0149 − 0.999i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.995 + 0.0896i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6768293146 - 0.7828886826i\)
\(L(\frac12)\) \(\approx\) \(0.6768293146 - 0.7828886826i\)
\(L(1)\) \(\approx\) \(0.8177139927 - 0.3019431262i\)
\(L(1)\) \(\approx\) \(0.8177139927 - 0.3019431262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.772 + 0.635i)T \)
3 \( 1 + (0.525 - 0.850i)T \)
5 \( 1 + (-0.280 - 0.959i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (0.0149 - 0.999i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.992 + 0.119i)T \)
41 \( 1 + (0.473 + 0.880i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.251 - 0.967i)T \)
53 \( 1 + (-0.873 - 0.486i)T \)
59 \( 1 + (-0.999 + 0.0299i)T \)
61 \( 1 + (-0.873 + 0.486i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.995 - 0.0896i)T \)
73 \( 1 + (0.251 + 0.967i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.936 - 0.351i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
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

−23.39172203937353658784707073537, −22.465701470442200681626236510257, −21.85380505616097335629817972106, −21.00085915266025029157542571771, −20.29788765854021713492309506779, −19.462653666086056068132419015564, −18.76005633115510347326024524206, −17.96389652006553453447786409436, −16.93415607194537169572452173132, −16.003867575725229470430898772060, −15.3372580813395208836253042248, −14.36665462315262341312980977446, −13.41972720720390276292909274105, −12.28457684174342455168792722859, −11.02084686480481776088690907416, −10.77018342248412704150748541943, −9.86819966080045359531211847449, −8.90326932800329325436561483039, −8.10736419936694914846432916124, −7.24955411384259353581407631570, −5.972265283064747583686399267696, −4.40151772997418704976961895085, −3.38092558723475557341445733054, −2.94438654731537455819482094673, −1.538414933746616381955768139026, 0.727771221024125182165853914477, 1.55184489476405169424563521059, 2.93317831126405081573035557072, 4.48641492752996554277804925402, 5.61732183027532323637471023735, 6.57248933946832867749809866815, 7.57977158891734378054730707324, 8.1606249482608579962725477671, 9.16441386198264906657958969626, 9.5527973669282875725660574031, 11.298668895486492285290481252772, 11.821882549369301688095787228183, 13.32257459295353268349443786853, 13.59466158115976169242195859156, 14.851776569022648042513489413855, 15.67172486075457773197977116705, 16.49450842522675670303263920255, 17.32310795944805557623087422108, 18.25332986501140006984116370579, 18.8300513251198173777390124084, 19.77935115718648724029027444625, 20.36521189748180132747522693453, 21.10607689137372847375091743615, 22.94038681929071272675570864339, 23.40031089182030758666186164386

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