Properties

Label 2-570-57.50-c1-0-18
Degree $2$
Conductor $570$
Sign $0.752 + 0.658i$
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 + (1.72 − 0.162i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.00 − 1.41i)6-s + 3.36·7-s + 0.999·8-s + (2.94 − 0.561i)9-s + (−0.866 − 0.499i)10-s + 0.795i·11-s + (−0.721 + 1.57i)12-s + (1.59 + 0.922i)13-s + (−1.68 − 2.91i)14-s + (1.41 − 1.00i)15-s + (−0.5 − 0.866i)16-s + (−6.17 + 3.56i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.995 − 0.0939i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.409 − 0.576i)6-s + 1.27·7-s + 0.353·8-s + (0.982 − 0.187i)9-s + (−0.273 − 0.158i)10-s + 0.239i·11-s + (−0.208 + 0.454i)12-s + (0.443 + 0.255i)13-s + (−0.449 − 0.779i)14-s + (0.364 − 0.259i)15-s + (−0.125 − 0.216i)16-s + (−1.49 + 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.752 + 0.658i$
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: \((2,\ 570,\ (\ :1/2),\ 0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87161 - 0.703617i\)
\(L(\frac12)\) \(\approx\) \(1.87161 - 0.703617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.162i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.831 - 4.27i)T \)
good7 \( 1 - 3.36T + 7T^{2} \)
11 \( 1 - 0.795iT - 11T^{2} \)
13 \( 1 + (-1.59 - 0.922i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.17 - 3.56i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.72 + 0.997i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.95 + 3.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.28iT - 31T^{2} \)
37 \( 1 - 3.63iT - 37T^{2} \)
41 \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.75 + 3.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.13 + 1.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.21 + 5.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.40 - 7.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.34 - 5.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.16 + 1.82i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.800 - 1.38i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.05 + 12.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 - 5.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 + (1.83 - 3.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.9 - 9.19i)T + (48.5 - 84.0i)T^{2} \)
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

−10.49868714340263291900600089510, −9.788023820595025926768552997333, −8.615455913254154124141377246366, −8.430781705911927868198038382838, −7.41288488589663101463087949991, −6.14325067033159692636221543608, −4.58613316196062775156976789821, −3.92054806319785561275818650516, −2.26606247165588683611625820309, −1.62251441672412676404855265629, 1.57243614971032717757109471937, 2.83074386570320039646389391408, 4.41119908739430887010625138167, 5.16666871274970020871814811313, 6.62349969169094531142969029117, 7.32755938385154938034405541086, 8.423698613179889148682671613953, 8.768583039123595938515145970375, 9.717927924702171499360374475326, 10.77640727770169291530274428981

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