Properties

Label 2-570-285.47-c1-0-28
Degree $2$
Conductor $570$
Sign $0.480 - 0.877i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + (0.819 + 0.573i)2-s + (1.50 + 0.855i)3-s + (0.342 + 0.939i)4-s + (1.97 − 1.04i)5-s + (0.743 + 1.56i)6-s + (−2.09 + 0.562i)7-s + (−0.258 + 0.965i)8-s + (1.53 + 2.57i)9-s + (2.21 + 0.282i)10-s + (2.25 − 1.30i)11-s + (−0.288 + 1.70i)12-s + (−2.88 + 0.252i)13-s + (−2.04 − 0.742i)14-s + (3.87 + 0.123i)15-s + (−0.766 + 0.642i)16-s + (1.92 + 1.34i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.869 + 0.493i)3-s + (0.171 + 0.469i)4-s + (0.884 − 0.465i)5-s + (0.303 + 0.638i)6-s + (−0.793 + 0.212i)7-s + (−0.0915 + 0.341i)8-s + (0.512 + 0.858i)9-s + (0.701 + 0.0892i)10-s + (0.679 − 0.392i)11-s + (−0.0832 + 0.493i)12-s + (−0.801 + 0.0700i)13-s + (−0.545 − 0.198i)14-s + (0.999 + 0.0320i)15-s + (−0.191 + 0.160i)16-s + (0.465 + 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.480 - 0.877i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43577 + 1.44358i\)
\(L(\frac12)\) \(\approx\) \(2.43577 + 1.44358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (-1.50 - 0.855i)T \)
5 \( 1 + (-1.97 + 1.04i)T \)
19 \( 1 + (-3.59 - 2.46i)T \)
good7 \( 1 + (2.09 - 0.562i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.25 + 1.30i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.88 - 0.252i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (-1.92 - 1.34i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (5.73 + 2.67i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.986 - 5.59i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.45 + 7.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.53 + 8.53i)T + 37iT^{2} \)
41 \( 1 + (3.23 + 3.86i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-7.06 + 3.29i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (-1.86 - 2.65i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (7.30 + 3.40i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.302 + 1.71i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.68 - 2.43i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.51 - 1.05i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (4.61 - 12.6i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.25 + 14.3i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (-7.19 - 8.57i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.93 + 1.85i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.349 - 0.293i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.74 - 2.48i)T + (-33.1 - 91.1i)T^{2} \)
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

−10.63630387911912181773988258494, −9.751851925078572162806105002815, −9.230373172929313786351555498960, −8.310454250236823503039985637626, −7.29705622287925794335586195555, −6.14826697690942505754480098899, −5.38248650145042078927120393561, −4.20613497349120786563186883738, −3.23459431344796938251495829538, −2.03776689283180431170308530933, 1.53161774258381377001692278083, 2.74931101228654801156853980126, 3.47593807575157898253752504444, 4.87503885947787488106672347469, 6.21047793930950795955488777366, 6.82923334567092311421209762694, 7.73189626091782085775257237377, 9.201243009268572241479334045747, 9.760478628799739763113485759723, 10.29115833436281668220018437514

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