Properties

Label 2-570-19.6-c1-0-10
Degree $2$
Conductor $570$
Sign $-0.996 + 0.0778i$
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  + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (−0.266 + 0.460i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−1.85 − 3.21i)11-s + (−0.499 + 0.866i)12-s + (−3.09 + 1.12i)13-s + (−0.0923 − 0.524i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.624 + 0.524i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (0.383 − 0.139i)6-s + (−0.100 + 0.174i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (−0.560 − 0.970i)11-s + (−0.144 + 0.250i)12-s + (−0.857 + 0.312i)13-s + (−0.0246 − 0.140i)14-s + (−0.0448 + 0.254i)15-s + (−0.234 − 0.0855i)16-s + (−0.151 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.996 + 0.0778i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.996 + 0.0778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000268658 - 0.00688996i\)
\(L(\frac12)\) \(\approx\) \(0.000268658 - 0.00688996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-4.21 - 1.10i)T \)
good7 \( 1 + (0.266 - 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.85 + 3.21i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.09 - 1.12i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.624 - 0.524i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.01 - 5.74i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.68 + 6.44i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.70 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.04T + 37T^{2} \)
41 \( 1 + (9.69 + 3.52i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.768 - 4.35i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.35 + 7.00i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.180 - 1.02i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-5.09 + 4.27i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.48 + 14.1i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.91 - 3.28i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.595 - 3.37i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.02 + 0.371i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (11.8 + 4.31i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (9.02 - 15.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.60 - 1.31i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-2.01 + 1.69i)T + (16.8 - 95.5i)T^{2} \)
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

−10.15763901174118781869631087998, −9.438459236863552819272322851923, −8.470974536317397295818123406750, −7.62315438788011425160192913827, −6.81010597487596057924132728797, −5.55963360788584169019642869555, −5.18795135548182688485425313032, −3.51384634605734658405478140339, −1.74049952061666223572130267527, −0.00477455418305308418209864698, 2.04360545762475382680193357773, 3.30996022686348678100707883742, 4.59771913846079886700333637162, 5.56811787935747050009170629895, 7.08277466405487647410146535229, 7.35597403532259004842622683381, 8.669762961442084297447980914466, 9.806395698795572577879211609853, 10.16542238661908827279510921629, 11.06905467709118763561140947525

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