Properties

Label 2-570-57.50-c1-0-5
Degree $2$
Conductor $570$
Sign $0.216 - 0.976i$
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.22 + 1.22i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (1.67 + 0.447i)6-s + 3.20·7-s + 0.999·8-s + (−0.00438 − 2.99i)9-s + (0.866 + 0.499i)10-s + 2.81i·11-s + (−0.449 − 1.67i)12-s + (−1.17 − 0.679i)13-s + (−1.60 − 2.77i)14-s + (0.447 − 1.67i)15-s + (−0.5 − 0.866i)16-s + (−2.48 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.706 + 0.707i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.683 + 0.182i)6-s + 1.21·7-s + 0.353·8-s + (−0.00146 − 0.999i)9-s + (0.273 + 0.158i)10-s + 0.849i·11-s + (−0.129 − 0.482i)12-s + (−0.326 − 0.188i)13-s + (−0.428 − 0.742i)14-s + (0.115 − 0.432i)15-s + (−0.125 − 0.216i)16-s + (−0.603 + 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.216 - 0.976i$
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: \((2,\ 570,\ (\ :1/2),\ 0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613951 + 0.492592i\)
\(L(\frac12)\) \(\approx\) \(0.613951 + 0.492592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-4.04 - 1.62i)T \)
good7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 - 2.81iT - 11T^{2} \)
13 \( 1 + (1.17 + 0.679i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.48 - 1.43i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.23 + 1.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.63 - 4.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.22 + 4.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.96 - 3.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.85 - 3.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.946 + 1.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.15 + 5.28i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.825 - 1.43i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.52 + 7.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.55 + 4.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 + (1.31 - 2.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.27 + 4.77i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88232493577198894176144095685, −10.23144888703532570969923992177, −9.443735436743535948939077357873, −8.351221379853054814865623363905, −7.55461933623460415505158856585, −6.37480622432997021792512347031, −4.92646432080829614525837827333, −4.49488661078651669220529780957, −3.17425916141695141310948872081, −1.51273267143605479134958370998, 0.58985163218622873406730941405, 2.08475626222573961386130171604, 4.20838630080507582716166550759, 5.26287879777080810868543081433, 5.90940603322796161257987717899, 7.20789817644333964960000413201, 7.70267783491724362132674239603, 8.520851468873239858553927966489, 9.496005862691503393716648986933, 10.87523503954780438506956271878

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