Properties

Label 2-570-95.64-c1-0-19
Degree $2$
Conductor $570$
Sign $0.225 + 0.974i$
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 + (2.12 + 0.710i)5-s + (0.499 − 0.866i)6-s − 4.67i·7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.19 − 0.445i)10-s − 3.96·11-s − 0.999i·12-s + (−0.698 − 0.403i)13-s + (−2.33 − 4.04i)14-s + (2.19 − 0.445i)15-s + (−0.5 − 0.866i)16-s + (−4.01 + 2.31i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.948 + 0.317i)5-s + (0.204 − 0.353i)6-s − 1.76i·7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.692 − 0.140i)10-s − 1.19·11-s − 0.288i·12-s + (−0.193 − 0.111i)13-s + (−0.624 − 1.08i)14-s + (0.565 − 0.114i)15-s + (−0.125 − 0.216i)16-s + (−0.974 + 0.562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.225 + 0.974i$
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: \((2,\ 570,\ (\ :1/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04872 - 1.62932i\)
\(L(\frac12)\) \(\approx\) \(2.04872 - 1.62932i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 0.710i)T \)
19 \( 1 + (-3.01 - 3.15i)T \)
good7 \( 1 + 4.67iT - 7T^{2} \)
11 \( 1 + 3.96T + 11T^{2} \)
13 \( 1 + (0.698 + 0.403i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.01 - 2.31i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.52 - 3.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.03 - 3.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 - 2.19iT - 37T^{2} \)
41 \( 1 + (2.02 + 3.51i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.36 + 3.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.12 - 4.69i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.34 - 0.778i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.20 + 3.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.42 + 3.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.08 - 1.87i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.2 - 5.91i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.67 + 8.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.92iT - 83T^{2} \)
89 \( 1 + (9.13 - 15.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 6.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48034841623663415971018151662, −10.05667767122654709986666001793, −8.915911986122514399404279290237, −7.52500126298863729135377182332, −7.08279150966468583334881363617, −5.90763680359452860721530266336, −4.83536315917141665983109980504, −3.67862178536922268212065046953, −2.64778911571576412377538435661, −1.31318759789367771285183895512, 2.44382145506491122270150127240, 2.71176097914444366493079247070, 4.71994373062196446173082365945, 5.27511591130516503419372165653, 6.11766769050665214129437340086, 7.27146122088299571268338063768, 8.527230131058721531143786686246, 9.008345920270243217825953611736, 9.828079684645541452500908426673, 11.00424616298702057754719885898

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