Properties

Label 2-570-19.11-c1-0-14
Degree $2$
Conductor $570$
Sign $-0.996 + 0.0841i$
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 − 2.64·7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 6.29·11-s − 0.999·12-s + (−1.32 + 2.29i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (2.82 − 4.88i)17-s − 0.999·18-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 − 0.999·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s − 1.89·11-s − 0.288·12-s + (−0.353 + 0.612i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.684 − 1.18i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.996 + 0.0841i$
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: \((2,\ 570,\ (\ :1/2),\ -0.996 + 0.0841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0495143 - 1.17540i\)
\(L(\frac12)\) \(\approx\) \(0.0495143 - 1.17540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 4.02i)T \)
good7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 + 6.29T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.82 + 4.88i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.82 - 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 - 8.29T + 37T^{2} \)
41 \( 1 + (-4.32 + 7.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.29 + 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.32 - 4.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.64 - 4.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.46 + 6.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.82 - 8.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.82 + 11.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.82 - 11.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.64 - 4.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + (6.61 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.822 + 1.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.21167479761409603935958184887, −9.653050966460284776950613624070, −8.633033013652491677984458721425, −7.66543134650304003649107453855, −6.62687841902815410737963705239, −5.54035984744261608199304680853, −4.68316212317298898606494462527, −3.07140812508194113775194924889, −2.48870576477034165880532921844, −0.54406922090901035924224570178, 2.60320832183567795665031102675, 3.47258175047305593146733615536, 4.66894962608126442547766659051, 5.83051844556322620255475760146, 6.38030463779941462824574333830, 7.88515414424776816259659230251, 8.116640656768927225030350494534, 9.632305621539471641616258867165, 10.11332518572573204153771165085, 10.89722673123411505526782246117

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