Properties

Label 2-570-19.11-c1-0-12
Degree $2$
Conductor $570$
Sign $-0.567 + 0.823i$
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 − 1.44·7-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 + (0.724 − 1.25i)14-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.775 − 1.34i)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 − 0.547·7-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 + (0.193 − 0.335i)14-s + (0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.188 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.567 + 0.823i$
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: \((2,\ 570,\ (\ :1/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176425 - 0.335625i\)
\(L(\frac12)\) \(\approx\) \(0.176425 - 0.335625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.17 + 1.25i)T \)
good7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.775 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.94 - 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.67 + 8.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.898T + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 + (2.17 - 3.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.89 + 10.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.550 - 0.953i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.72 - 6.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.22 + 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.67 - 6.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.67 + 2.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.22 - 9.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.44 - 5.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-2.72 - 4.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.22 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.27513628734788854988141662902, −9.523076555515430079391525707826, −8.460283118625988891766630137652, −7.62440058200120831370906751720, −7.10580364137254428817306725166, −5.97681808692294597622231134292, −5.10916755571389116293082246517, −3.50442520325816704486954840895, −2.36383698899111400902639438441, −0.21731006135595366352541085317, 2.00783485135530071835925684591, 3.22182525093087833679656898344, 4.30379635883708533493032749221, 5.19595966259254896273159400494, 6.67663650933266709147450144050, 7.69473949047785222056410605547, 8.772184325426869532661805325246, 9.189139931182407019854089138941, 10.31396129336690662046538138710, 10.72925604717872236559527821837

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