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  + 0.879·2-s − 1.22·4-s − 1.34·5-s + 7-s − 2.83·8-s − 1.18·10-s − 1.65·11-s − 3.36·13-s + 0.879·14-s − 0.0418·16-s − 0.467·17-s − 3.22·19-s + 1.65·20-s − 1.45·22-s − 8.94·23-s − 3.18·25-s − 2.96·26-s − 1.22·28-s − 6.26·29-s + 9.23·31-s + 5.63·32-s − 0.411·34-s − 1.34·35-s + 9.23·37-s − 2.83·38-s + 3.82·40-s − 3.41·41-s + ⋯
L(s)  = 1  + 0.621·2-s − 0.613·4-s − 0.602·5-s + 0.377·7-s − 1.00·8-s − 0.374·10-s − 0.498·11-s − 0.934·13-s + 0.235·14-s − 0.0104·16-s − 0.113·17-s − 0.740·19-s + 0.369·20-s − 0.309·22-s − 1.86·23-s − 0.636·25-s − 0.581·26-s − 0.231·28-s − 1.16·29-s + 1.65·31-s + 0.996·32-s − 0.0705·34-s − 0.227·35-s + 1.51·37-s − 0.460·38-s + 0.604·40-s − 0.532·41-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 \( 1 - 0.879T + 2T^{2} \)
5 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 + 1.65T + 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 + 3.22T + 19T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 - 9.23T + 31T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 + 4.41T + 43T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 - 0.596T + 67T^{2} \)
71 \( 1 - 0.554T + 71T^{2} \)
73 \( 1 - 2.04T + 73T^{2} \)
79 \( 1 + 2.40T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + 9.08T + 89T^{2} \)
97 \( 1 + 1.89T + 97T^{2} \)
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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

−10.18353691824275963884012225753, −9.572310937334592632146804226006, −8.245839697179878784779077894000, −7.913953997702209406289956428163, −6.50057767771042358522708295707, −5.46011383056055644730047054974, −4.53132672795117656662485945350, −3.80709206323046140474176002180, −2.38454120518082410602072645616, 0, 2.38454120518082410602072645616, 3.80709206323046140474176002180, 4.53132672795117656662485945350, 5.46011383056055644730047054974, 6.50057767771042358522708295707, 7.913953997702209406289956428163, 8.245839697179878784779077894000, 9.572310937334592632146804226006, 10.18353691824275963884012225753

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