Properties

Degree $2$
Conductor $567$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.239·2-s − 1.94·4-s + 1.18·5-s − 7-s + 0.942·8-s − 0.282·10-s − 3.70·11-s + 13-s + 0.239·14-s + 3.66·16-s − 6.94·17-s + 1.94·19-s − 2.29·20-s + 0.885·22-s − 5.60·23-s − 3.60·25-s − 0.239·26-s + 1.94·28-s + 0.239·29-s + 1.66·31-s − 2.76·32-s + 1.66·34-s − 1.18·35-s − 9.54·37-s − 0.464·38-s + 1.11·40-s − 10.1·41-s + ⋯
L(s)  = 1  − 0.169·2-s − 0.971·4-s + 0.528·5-s − 0.377·7-s + 0.333·8-s − 0.0893·10-s − 1.11·11-s + 0.277·13-s + 0.0639·14-s + 0.915·16-s − 1.68·17-s + 0.445·19-s − 0.513·20-s + 0.188·22-s − 1.16·23-s − 0.720·25-s − 0.0468·26-s + 0.367·28-s + 0.0444·29-s + 0.298·31-s − 0.488·32-s + 0.284·34-s − 0.199·35-s − 1.56·37-s − 0.0753·38-s + 0.176·40-s − 1.59·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{567} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 0.239T + 2T^{2} \)
5 \( 1 - 1.18T + 5T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + 5.60T + 23T^{2} \)
29 \( 1 - 0.239T + 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 2.22T + 43T^{2} \)
47 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 2.60T + 59T^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 7.37T + 79T^{2} \)
83 \( 1 + 6.94T + 83T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 - 7.16T + 97T^{2} \)
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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

−10.16751574014947657595814419137, −9.478926114819564418637865086677, −8.611894876496353828936787676140, −7.85211843023724863786930854203, −6.60980171432291819252986663250, −5.58760781409360007139260328445, −4.71364151121870565129588476827, −3.55591448621153857250248519602, −2.07072477954214546651559024617, 0, 2.07072477954214546651559024617, 3.55591448621153857250248519602, 4.71364151121870565129588476827, 5.58760781409360007139260328445, 6.60980171432291819252986663250, 7.85211843023724863786930854203, 8.611894876496353828936787676140, 9.478926114819564418637865086677, 10.16751574014947657595814419137

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