Properties

Label 2-570-19.11-c1-0-3
Degree $2$
Conductor $570$
Sign $0.915 - 0.401i$
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + 3.77·11-s + 0.999·12-s + (2.38 + 4.13i)13-s + (−0.5 + 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-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 − 0.377·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + 1.13·11-s + 0.288·12-s + (0.661 + 1.14i)13-s + (−0.133 + 0.231i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.915 - 0.401i$
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: \((2,\ 570,\ (\ :1/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44099 + 0.301765i\)
\(L(\frac12)\) \(\approx\) \(1.44099 + 0.301765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.88 - 3.26i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3.77T + 11T^{2} \)
13 \( 1 + (-2.38 - 4.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.88 - 3.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.88 + 3.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.38 - 7.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.88 + 8.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.77 + 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.38 + 2.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.38 - 9.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.77 + 6.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.61 + 2.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.15 - 8.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (8.65 + 14.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.04774909708136299323012228940, −9.892622392004703816488616237977, −9.395698081598537866653099355433, −8.387877012469351659656048114021, −6.86685834437280169404905797608, −6.25294014105870489654570349695, −5.02278075545527508646029796417, −3.92043005706454492913084453699, −3.27889352658067559263446694536, −1.49379574705798438540401070417, 0.897036902763835848930715897183, 2.95744198429218754653464790257, 4.15240873729066086379028745080, 5.26059112806970958328214933517, 6.21572377067093524337183460088, 6.90386483072044435145287749766, 7.952607136391824461827559761158, 8.694965034724487515105278609053, 9.642127164153768272063868732509, 10.82540974983481333471424884927

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