↑

Properties

Label 2-567-7.2-c1-0-6
Degree $2$
Conductor $567$
Sign $-0.507 - 0.861i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  + (−0.335 + 0.580i)2-s + (0.775 + 1.34i)4-s + (−0.712 + 1.23i)5-s + (0.145 − 2.64i)7-s − 2.38·8-s + (−0.477 − 0.827i)10-s + (2.46 + 4.27i)11-s + 2.75·13-s + (1.48 + 0.969i)14-s + (−0.752 + 1.30i)16-s + (0.559 + 0.969i)17-s + (−2.00 + 3.47i)19-s − 2.20·20-s − 3.30·22-s + (−2.71 + 4.70i)23-s + ⋯
L(s)  = 1  + (−0.236 + 0.410i)2-s + (0.387 + 0.671i)4-s + (−0.318 + 0.551i)5-s + (0.0548 − 0.998i)7-s − 0.841·8-s + (−0.151 − 0.261i)10-s + (0.743 + 1.28i)11-s + 0.763·13-s + (0.396 + 0.259i)14-s + (−0.188 + 0.326i)16-s + (0.135 + 0.235i)17-s + (−0.460 + 0.797i)19-s − 0.494·20-s − 0.704·22-s + (−0.566 + 0.981i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607631 + 1.06285i\)
\(L(\frac12)\) \(\approx\) \(0.607631 + 1.06285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.145 + 2.64i)T \)
good2 \( 1 + (0.335 - 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.712 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.46 - 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.00 - 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.81T + 29T^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.248T + 41T^{2} \)
43 \( 1 - 0.996T + 43T^{2} \)
47 \( 1 + (-4.73 + 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.29 - 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0376 - 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + (-6.76 + 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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

−11.10614520086854398322998315597, −10.16143788433812712159784692084, −9.237324865806254912827842421823, −8.129612387123195935033176094466, −7.33035818051382973174573408381, −6.90372116107449459770496069071, −5.80959712125134745157424283457, −4.03095106806893397634972566071, −3.61097716429928146524930311973, −1.85387515140501107271966496794, 0.76678249567474272967826071949, 2.22830709646126932455479883682, 3.48455210657357066654789028641, 4.91802018052719322523965348344, 5.98439965266830072880145648090, 6.49759294803204512306296840035, 8.127259308853106792637429659022, 8.916087877731068824048758145699, 9.342341210490300429543105339439, 10.72497844154274829547604820464

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