Properties

Label 2-570-285.179-c1-0-36
Degree $2$
Conductor $570$
Sign $-0.419 + 0.907i$
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 + (1.15 − 1.29i)3-s + (0.499 + 0.866i)4-s + (2.17 − 0.520i)5-s + (−1.64 + 0.540i)6-s − 4.14i·7-s − 0.999i·8-s + (−0.333 − 2.98i)9-s + (−2.14 − 0.636i)10-s + 0.276i·11-s + (1.69 + 0.354i)12-s + (0.418 + 0.725i)13-s + (−2.07 + 3.58i)14-s + (1.83 − 3.40i)15-s + (−0.5 + 0.866i)16-s + (−2.82 + 4.90i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.666 − 0.745i)3-s + (0.249 + 0.433i)4-s + (0.972 − 0.232i)5-s + (−0.671 + 0.220i)6-s − 1.56i·7-s − 0.353i·8-s + (−0.111 − 0.993i)9-s + (−0.677 − 0.201i)10-s + 0.0834i·11-s + (0.489 + 0.102i)12-s + (0.116 + 0.201i)13-s + (−0.553 + 0.958i)14-s + (0.474 − 0.880i)15-s + (−0.125 + 0.216i)16-s + (−0.686 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.419 + 0.907i$
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: \((2,\ 570,\ (\ :1/2),\ -0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823068 - 1.28688i\)
\(L(\frac12)\) \(\approx\) \(0.823068 - 1.28688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 1.29i)T \)
5 \( 1 + (-2.17 + 0.520i)T \)
19 \( 1 + (2.04 + 3.85i)T \)
good7 \( 1 + 4.14iT - 7T^{2} \)
11 \( 1 - 0.276iT - 11T^{2} \)
13 \( 1 + (-0.418 - 0.725i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.82 - 4.90i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.101 - 0.176i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.06 - 7.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.54iT - 31T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + (-2.91 + 5.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.67 - 5.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.89 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.06 - 4.65i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.94 - 3.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.56 + 4.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.36 + 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.57 - 3.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-14.0 - 8.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.50T + 83T^{2} \)
89 \( 1 + (-3.27 - 5.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.24 + 5.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.55838467246098523228243157442, −9.442575988046434248090512543005, −8.799901025303188884236353254162, −7.898161533776881038956289988041, −6.87911458611010848877798919932, −6.39928581156767598093372008755, −4.60050110441494357825452376297, −3.40737660087330331865866188280, −2.06076095083955816509033802229, −1.03175590049426425026115341260, 2.15844320746517035512926350425, 2.81077895105604000379398364002, 4.62277071632049389950410998413, 5.68438819306209677830473334567, 6.30200188556263214734606838906, 7.77686357375553597352580842396, 8.570377035210503109295555152652, 9.426005185786779354358253474069, 9.670320151647384155598324027896, 10.77640575491276625345678176902

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