Properties

Label 2-570-19.11-c1-0-0
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 + (−1.67 + 2.90i)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 + (−0.407 + 0.705i)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

\(L(1)\) \(\approx\) \(0.0748582 + 0.168909i\)
\(L(\frac12)\) \(\approx\) \(0.0748582 + 0.168909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 2.90i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.67 + 8.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-3.17 + 5.50i)T + (-20.5 - 35.5i)T^{2} \)
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 \( 1 + (6.35 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.67 + 4.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 + 9.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.03 + 8.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.32 - 4.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.35 + 4.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (0.179 + 0.310i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.03 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78597656144941086213255137445, −10.41860039317908647866050457393, −9.293337254533572796871403405930, −9.001306323609548345369321401063, −7.38241732797961059606230005261, −6.08073476862071474196335228095, −5.67232768690701720642479908622, −4.18111194001153863925359685124, −3.53901899644976064063881312067, −2.08740078002895903077963268781, 0.089032823645111286563519576607, 2.65727370149076214949415935096, 3.49853706084626224085784772319, 5.15533378634657451355223621147, 5.90690892526157251968392702098, 6.84818255786111597811264061326, 7.29974219334851439077275392342, 8.596536613407218102778969741490, 9.438028183305060786889979548674, 10.52325981500705287935842160499

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