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Properties

Label 2-570-95.64-c1-0-9
Degree $2$
Conductor $570$
Sign $0.974 + 0.226i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.118 + 2.23i)5-s + (−0.499 + 0.866i)6-s − 2.79i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1.01 + 1.99i)10-s + 4.02·11-s + 0.999i·12-s + (0.0960 + 0.0554i)13-s + (−1.39 − 2.42i)14-s + (−1.01 − 1.99i)15-s + (−0.5 − 0.866i)16-s + (3.68 − 2.12i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.0531 + 0.998i)5-s + (−0.204 + 0.353i)6-s − 1.05i·7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.320 + 0.630i)10-s + 1.21·11-s + 0.288i·12-s + (0.0266 + 0.0153i)13-s + (−0.373 − 0.647i)14-s + (−0.261 − 0.514i)15-s + (−0.125 − 0.216i)16-s + (0.893 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88523 - 0.216199i\)
\(L(\frac12)\) \(\approx\) \(1.88523 - 0.216199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.118 - 2.23i)T \)
19 \( 1 + (-0.163 - 4.35i)T \)
good7 \( 1 + 2.79iT - 7T^{2} \)
11 \( 1 - 4.02T + 11T^{2} \)
13 \( 1 + (-0.0960 - 0.0554i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.68 + 2.12i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.65 - 4.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.907 + 1.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 + 1.68iT - 37T^{2} \)
41 \( 1 + (3.88 + 6.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.46 - 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.17 + 2.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.45 + 3.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.188 + 0.326i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.18 - 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.82 + 2.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.90 + 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.33 + 4.81i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.11 + 3.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.25iT - 83T^{2} \)
89 \( 1 + (-5.01 + 8.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.52 - 2.61i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.71625828599372864566531545833, −10.15404104525744504051311352365, −9.353663556196018757250292132345, −7.72289148495350450014811334687, −6.87908868001913821517042770345, −6.19387037856450769913303118099, −4.99366976425872956300024037353, −3.86095180823226042650136253781, −3.21705453215433387644003661503, −1.28627081245362057779957284341, 1.33117195892030511643172331287, 3.00440261228411149434321571303, 4.51489472088351161727381127967, 5.16042885154862421967675080139, 6.15925335506142758778176162938, 6.85858502647041137181352350009, 8.219302451891734694762857945740, 8.809493324568664771916502024539, 9.739902261063387151350436698518, 11.16679826911737388758869488529

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