Properties

Label 2-570-19.11-c1-0-13
Degree $2$
Conductor $570$
Sign $-0.999 + 0.0128i$
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 − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s − 4.77·11-s + 0.999·12-s + (−1.88 − 3.26i)13-s + (−0.5 + 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-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 − 0.377·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s − 1.43·11-s + 0.288·12-s + (−0.523 − 0.906i)13-s + (−0.133 + 0.231i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.999 + 0.0128i$
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: \((2,\ 570,\ (\ :1/2),\ -0.999 + 0.0128i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00179703 - 0.279854i\)
\(L(\frac12)\) \(\approx\) \(0.00179703 - 0.279854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 + 4.13i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 + (1.88 + 3.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.38 + 4.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (2.38 - 4.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.113 - 0.197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.613 + 1.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.77 - 3.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.88 - 4.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.11 - 1.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.77 - 8.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.88 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.65 + 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-4.15 - 7.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.52957039612211570109706544884, −9.796554957946411638354945266163, −8.678955849365812355778499503767, −7.63692490956924695363476006758, −6.50482067623827917410427134437, −5.37490950381912048044220657753, −4.67675488311901983742177546034, −3.32721916202657845987984950242, −2.53392617694469497759118158751, −0.13528159176385247344993831928, 2.14941285181559990841856776590, 3.67021370753673792079380701801, 4.87892045185958998425650059452, 5.68611338796817555120309423138, 6.64390014301255089113725834474, 7.63486540773088237729388052755, 8.160279572286393682402300817129, 9.342046680625294717729957414009, 10.25184795248686885223432368145, 11.37382755973769663419724896702

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