Properties

Label 2-567-7.6-c0-0-3
Degree $2$
Conductor $567$
Sign $1$
Analytic cond. $0.282969$
Root an. cond. $0.531949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 1.99·4-s − 7-s + 1.73·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s − 3.46·44-s + 49-s + 1.73·50-s − 1.73·53-s − 1.73·56-s − 1.00·64-s − 67-s + 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 1.73·98-s + 1.99·100-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s − 7-s + 1.73·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s − 3.46·44-s + 49-s + 1.73·50-s − 1.73·53-s − 1.73·56-s − 1.00·64-s − 67-s + 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 1.73·98-s + 1.99·100-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (244, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.958716653\)
\(L(\frac12)\) \(\approx\) \(1.958716653\)
\(L(1)\) 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 \( 1 + T \)
good2 \( 1 - 1.73T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.07067185158620858047463607831, −10.46569589200719895761701231757, −9.371246171645520668549116380100, −7.970835435007244695063973068796, −7.02666353442432348798601254878, −6.12269619030713511372617528053, −5.33569885850940032010161472943, −4.42938801666431015361494420647, −3.20497650277098718036239044932, −2.52899724031951772135360380788, 2.52899724031951772135360380788, 3.20497650277098718036239044932, 4.42938801666431015361494420647, 5.33569885850940032010161472943, 6.12269619030713511372617528053, 7.02666353442432348798601254878, 7.970835435007244695063973068796, 9.371246171645520668549116380100, 10.46569589200719895761701231757, 11.07067185158620858047463607831

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