↑

Properties

Label 2-570-19.7-c3-0-33
Degree $2$
Conductor $570$
Sign $-0.933 + 0.357i$
Analytic cond. $33.6310$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s + (3 − 5.19i)6-s − 10.6·7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (5 − 8.66i)10-s − 15.3·11-s − 12·12-s + (23 − 39.8i)13-s + (10.6 + 18.3i)14-s + (−7.50 + 12.9i)15-s + (−8 − 13.8i)16-s + (−11.5 − 19.9i)17-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 − 0.573·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s − 0.419·11-s − 0.288·12-s + (0.490 − 0.849i)13-s + (0.202 + 0.351i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.164 − 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.357i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.933 + 0.357i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ -0.933 + 0.357i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2478314540\)
\(L(\frac12)\) \(\approx\) \(0.2478314540\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
19 \( 1 + (-4.56 - 82.6i)T \)
good7 \( 1 + 10.6T + 343T^{2} \)
11 \( 1 + 15.3T + 1.33e3T^{2} \)
13 \( 1 + (-23 + 39.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (11.5 + 19.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (92.4 - 160. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-111. + 193. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 32.5T + 2.97e4T^{2} \)
37 \( 1 + 310.T + 5.06e4T^{2} \)
41 \( 1 + (174. + 302. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (92.2 + 159. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (112. - 194. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-226. + 392. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-186. - 323. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (28.0 - 48.5i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-372. + 645. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (530. + 918. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (426. + 738. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (479. + 830. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 205.T + 5.71e5T^{2} \)
89 \( 1 + (-183. + 317. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (193. + 335. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−10.20372201616974401849583039278, −9.286550151990069964190005523229, −8.291295336502624585711386085226, −7.54053354164858673114938632501, −6.21015696420021900240581739986, −5.23833419595007916557684371139, −3.77786643214017824534297637081, −3.13256302532425347209460825473, −1.87402130329706119333328002073, −0.079380417476605580142406197661, 1.37012071921601830722976617882, 2.73632802163816079456265817738, 4.24230862582467629308043377330, 5.35030647327690482175736860261, 6.58143921318219377394820655569, 6.86643847129048890658662568116, 8.336433590377715282117469354483, 8.656357527892462949005016823275, 9.674790327407493354218857324324, 10.48396227301231886018105884839

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