Properties

Label 2-570-19.11-c1-0-7
Degree $2$
Conductor $570$
Sign $0.671 + 0.740i$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s + 4.35·7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 3·11-s + 0.999·12-s + (2 + 3.46i)13-s + (2.17 − 3.77i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (2.67 − 4.64i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + 1.64·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s − 0.904·11-s + 0.288·12-s + (0.554 + 0.960i)13-s + (0.582 − 1.00i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.649 − 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70092 - 0.753822i\)
\(L(\frac12)\) \(\approx\) \(1.70092 - 0.753822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-2.17 + 3.77i)T \)
good7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.67 + 4.64i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.320 + 0.555i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.71T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (1.17 - 2.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.53 + 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.35 + 4.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.67 - 2.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.32 + 2.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.03 - 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.67 - 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-4.17 - 7.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.03 - 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82655679927527048562359526544, −9.889727629942009419294603857492, −9.010065911134913108008138508645, −8.160567553500970287097020681363, −7.01502423702084290258210777783, −5.40540472116662672330317183893, −5.07083348932359159497437956352, −4.12981803448696506781788778313, −2.62528540222150442095753642479, −1.23427886987856013288839635268, 1.50848181995704059546899984299, 3.05096724082480074397578310433, 4.53924983221258163860377843305, 5.51709216339812757143367484910, 6.07795267019598013796138337880, 7.48364351778155417051606713270, 7.955996215159635187428672572197, 8.617919955832661087063248857408, 10.35915695807569511781551493274, 10.73412271768726980452048389689

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