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Properties

Label 2-570-285.179-c1-0-13
Degree $2$
Conductor $570$
Sign $0.844 + 0.535i$
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 + (−1.69 − 0.359i)3-s + (0.499 + 0.866i)4-s + (−0.161 − 2.23i)5-s + (1.28 + 1.15i)6-s + 1.18i·7-s − 0.999i·8-s + (2.74 + 1.21i)9-s + (−0.975 + 2.01i)10-s + 4.89i·11-s + (−0.536 − 1.64i)12-s + (−0.622 − 1.07i)13-s + (0.591 − 1.02i)14-s + (−0.527 + 3.83i)15-s + (−0.5 + 0.866i)16-s + (0.318 − 0.552i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.978 − 0.207i)3-s + (0.249 + 0.433i)4-s + (−0.0721 − 0.997i)5-s + (0.525 + 0.472i)6-s + 0.446i·7-s − 0.353i·8-s + (0.913 + 0.405i)9-s + (−0.308 + 0.636i)10-s + 1.47i·11-s + (−0.154 − 0.475i)12-s + (−0.172 − 0.299i)13-s + (0.157 − 0.273i)14-s + (−0.136 + 0.990i)15-s + (−0.125 + 0.216i)16-s + (0.0773 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735429 - 0.213558i\)
\(L(\frac12)\) \(\approx\) \(0.735429 - 0.213558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.359i)T \)
5 \( 1 + (0.161 + 2.23i)T \)
19 \( 1 + (-4.35 - 0.0120i)T \)
good7 \( 1 - 1.18iT - 7T^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 + (0.622 + 1.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.318 + 0.552i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.859 - 1.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.94 - 3.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.45iT - 31T^{2} \)
37 \( 1 - 9.66T + 37T^{2} \)
41 \( 1 + (-6.08 + 10.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.32 - 5.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0980 - 0.169i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.57 - 0.908i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.25 - 2.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.71 + 9.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.67 - 4.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.99 + 5.18i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.20 + 4.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.74 - 3.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.21T + 83T^{2} \)
89 \( 1 + (-2.46 - 4.26i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.141 - 0.245i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.69608080290651469874574114883, −9.596389119519186806514958297458, −9.306921796790696268545918402423, −7.81419421433091536064288305126, −7.36376099953889745932737995531, −6.00781833117190203113770425412, −5.08840982896476633107948486874, −4.19270391076113954462595086584, −2.25250658838551459354781640363, −0.941541134858459114332737137855, 0.886790533708976558045905637504, 2.96313357977824257453288990513, 4.26239672839519262020118092726, 5.66391184491003906161140037617, 6.28854715549561126793939091066, 7.16223206367638418332615898676, 7.964404107141995352244857203525, 9.240114539792739432183454288074, 10.07771704335931845269797537319, 10.87603328606301654587193332156

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