Properties

Label 2-570-19.11-c1-0-4
Degree $2$
Conductor $570$
Sign $0.875 - 0.483i$
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 + 3.44·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 3·11-s − 0.999·12-s + (2.44 + 4.24i)13-s + (−1.72 + 2.98i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (3.22 − 5.58i)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 + 1.30·7-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 + (0.679 + 1.17i)13-s + (−0.460 + 0.798i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.782 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.875 - 0.483i$
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: \((2,\ 570,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40393 + 0.362205i\)
\(L(\frac12)\) \(\approx\) \(1.40393 + 0.362205i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.17 - 2.98i)T \)
good7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.94 + 3.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.67 - 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 + 0.101T + 37T^{2} \)
41 \( 1 + (-5.17 + 8.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.44 - 9.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.27 - 2.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.77 + 6.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.67 + 6.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.67 - 9.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.77 - 4.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.44 + 2.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.79T + 83T^{2} \)
89 \( 1 + (-0.275 - 0.476i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.77 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.81623345064611002952414967602, −9.820857940087989564766594802812, −8.773891085615086353997086353264, −7.970542525834146216241793878092, −7.47026537422130300335110121314, −6.47721109688871474794801523110, −5.35255967979779837355835584778, −4.37973104396228211221740234972, −2.75565949865289139121808060381, −1.29917329398976417970806965830, 1.19695066608824315690095506392, 2.71529160883312091550446797708, 3.88434408569891315821077984489, 4.92197487259708444174753577423, 5.75584529183925547781741132149, 7.75446987128051780598462143209, 8.075686864423794738016423408605, 8.781689202754661752418025588329, 10.12506361320116839230826489852, 10.45655877382274227681598281962

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