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Properties

Label 2-570-57.8-c1-0-13
Degree $2$
Conductor $570$
Sign $0.883 - 0.468i$
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 + (1.62 + 0.593i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (−1.32 + 1.11i)6-s − 0.387·7-s + 0.999·8-s + (2.29 + 1.93i)9-s + (0.866 − 0.499i)10-s − 6.28i·11-s + (−0.299 − 1.70i)12-s + (5.96 − 3.44i)13-s + (0.193 − 0.335i)14-s + (−1.11 − 1.32i)15-s + (−0.5 + 0.866i)16-s + (4.63 + 2.67i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.939 + 0.342i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.542 + 0.453i)6-s − 0.146·7-s + 0.353·8-s + (0.764 + 0.644i)9-s + (0.273 − 0.158i)10-s − 1.89i·11-s + (−0.0863 − 0.492i)12-s + (1.65 − 0.954i)13-s + (0.0517 − 0.0897i)14-s + (−0.287 − 0.342i)15-s + (−0.125 + 0.216i)16-s + (1.12 + 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59403 + 0.396824i\)
\(L(\frac12)\) \(\approx\) \(1.59403 + 0.396824i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.593i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.936 - 4.25i)T \)
good7 \( 1 + 0.387T + 7T^{2} \)
11 \( 1 + 6.28iT - 11T^{2} \)
13 \( 1 + (-5.96 + 3.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.63 - 2.67i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.57 - 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 - 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.87iT - 31T^{2} \)
37 \( 1 + 2.54iT - 37T^{2} \)
41 \( 1 + (1.40 - 2.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.588 - 1.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.74 + 3.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.97 + 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.556 + 0.964i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 2.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.95 + 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.17 - 7.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.890 - 1.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.3 + 7.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.22iT - 83T^{2} \)
89 \( 1 + (-7.49 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.33 + 3.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.56398608706534979406984143030, −9.836330857294482229176114143035, −8.649338151078952843783840620687, −8.264184332372645579662387291472, −7.71084774207705086785044741060, −6.08741359805818242757039765748, −5.57245283675567276040715671550, −3.78556841542523659622118102358, −3.37435655195601676383937468481, −1.20853528451391728397738174071, 1.44668314096905947660447100710, 2.61032102996064243539029184027, 3.76874449306306771229223643987, 4.59438957199348568853893404135, 6.51770383110763048581005769858, 7.26870123713830168968349424807, 8.114627005298514819370130863595, 9.008974013364641692737030195320, 9.727100754241175148722150288772, 10.47590836392881365697652224178

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