Properties

Label 2-570-57.50-c1-0-2
Degree $2$
Conductor $570$
Sign $0.879 - 0.476i$
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 + (−0.929 − 1.46i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.800 + 1.53i)6-s − 4.66·7-s + 0.999·8-s + (−1.27 + 2.71i)9-s + (−0.866 − 0.499i)10-s + 3.04i·11-s + (1.73 − 0.0746i)12-s + (3.56 + 2.06i)13-s + (2.33 + 4.03i)14-s + (−1.53 − 0.800i)15-s + (−0.5 − 0.866i)16-s + (2.22 − 1.28i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.536 − 0.843i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.326 + 0.627i)6-s − 1.76·7-s + 0.353·8-s + (−0.423 + 0.905i)9-s + (−0.273 − 0.158i)10-s + 0.918i·11-s + (0.499 − 0.0215i)12-s + (0.989 + 0.571i)13-s + (0.623 + 1.07i)14-s + (−0.396 − 0.206i)15-s + (−0.125 − 0.216i)16-s + (0.540 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.879 - 0.476i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.879 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.532666 + 0.134949i\)
\(L(\frac12)\) \(\approx\) \(0.532666 + 0.134949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.929 + 1.46i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (3.71 - 2.28i)T \)
good7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 - 3.04iT - 11T^{2} \)
13 \( 1 + (-3.56 - 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.22 + 1.28i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.586 - 0.338i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.992 - 1.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.29iT - 31T^{2} \)
37 \( 1 - 4.49iT - 37T^{2} \)
41 \( 1 + (1.65 + 2.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.86 - 4.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.58 + 4.38i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.57 + 7.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.02 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.95 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.56 + 1.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.25 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.18 - 3.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.74 + 3.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.61iT - 83T^{2} \)
89 \( 1 + (4.81 - 8.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.00 - 4.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68272816584075284708500116291, −10.02696956610190944499374508587, −9.207294596196314806847943699604, −8.285403045562618443601897989430, −6.98337207582443097233511270520, −6.49304971993288306810193751148, −5.42333616367857184422812771664, −3.93796225826731732819519401925, −2.66879919818314517513402980900, −1.36650031489126998120665120923, 0.39888781036296258536208895218, 3.07693770383093901301011791145, 3.93501503452904525322720070021, 5.50792587104026833006201032117, 6.15717393372777525673292326380, 6.63993403644949660153111223920, 8.165971887529308245789803691836, 9.156905293482291887309347839583, 9.674659077994716705028471870305, 10.58916698719419808171027843651

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