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  − 1.34·2-s − 0.184·4-s − 2.53·5-s + 7-s + 2.94·8-s + 3.41·10-s − 0.467·11-s + 5.82·13-s − 1.34·14-s − 3.59·16-s − 3.87·17-s − 2.18·19-s + 0.467·20-s + 0.630·22-s + 0.106·23-s + 1.41·25-s − 7.84·26-s − 0.184·28-s − 8.78·29-s − 7.68·31-s − 1.04·32-s + 5.22·34-s − 2.53·35-s − 7.68·37-s + 2.94·38-s − 7.45·40-s + 2.22·41-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s − 1.13·5-s + 0.377·7-s + 1.04·8-s + 1.07·10-s − 0.141·11-s + 1.61·13-s − 0.360·14-s − 0.899·16-s − 0.940·17-s − 0.501·19-s + 0.104·20-s + 0.134·22-s + 0.0221·23-s + 0.282·25-s − 1.53·26-s − 0.0349·28-s − 1.63·29-s − 1.37·31-s − 0.184·32-s + 0.896·34-s − 0.428·35-s − 1.26·37-s + 0.477·38-s − 1.17·40-s + 0.347·41-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 \( 1 + 1.34T + 2T^{2} \)
5 \( 1 + 2.53T + 5T^{2} \)
11 \( 1 + 0.467T + 11T^{2} \)
13 \( 1 - 5.82T + 13T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 - 0.106T + 23T^{2} \)
29 \( 1 + 8.78T + 29T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 1.22T + 43T^{2} \)
47 \( 1 - 5.33T + 47T^{2} \)
53 \( 1 - 0.716T + 53T^{2} \)
59 \( 1 + 0.736T + 59T^{2} \)
61 \( 1 - 0.958T + 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 2.73T + 83T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−10.46909880824011790021626696102, −8.983654195024787525592099519702, −8.760614867739603285440135304412, −7.78014400988830651791169088806, −7.13735710448351875663360313624, −5.74036438185373714314048704932, −4.37809598544036147863262519473, −3.69179883501881382005233149482, −1.67718686743966778762820678462, 0, 1.67718686743966778762820678462, 3.69179883501881382005233149482, 4.37809598544036147863262519473, 5.74036438185373714314048704932, 7.13735710448351875663360313624, 7.78014400988830651791169088806, 8.760614867739603285440135304412, 8.983654195024787525592099519702, 10.46909880824011790021626696102

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