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  − 2.46·2-s + 4.05·4-s − 2.59·5-s − 7-s − 5.05·8-s + 6.38·10-s + 4.51·11-s + 13-s + 2.46·14-s + 4.32·16-s − 0.945·17-s − 4.05·19-s − 10.5·20-s − 11.1·22-s − 0.273·23-s + 1.72·25-s − 2.46·26-s − 4.05·28-s + 2.46·29-s + 2.32·31-s − 0.539·32-s + 2.32·34-s + 2.59·35-s + 1.78·37-s + 9.97·38-s + 13.1·40-s − 6.40·41-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.02·4-s − 1.15·5-s − 0.377·7-s − 1.78·8-s + 2.01·10-s + 1.36·11-s + 0.277·13-s + 0.657·14-s + 1.08·16-s − 0.229·17-s − 0.930·19-s − 2.35·20-s − 2.36·22-s − 0.0569·23-s + 0.345·25-s − 0.482·26-s − 0.766·28-s + 0.456·29-s + 0.418·31-s − 0.0953·32-s + 0.399·34-s + 0.438·35-s + 0.292·37-s + 1.61·38-s + 2.07·40-s − 1.00·41-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 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 0.945T + 17T^{2} \)
19 \( 1 + 4.05T + 19T^{2} \)
23 \( 1 + 0.273T + 23T^{2} \)
29 \( 1 - 2.46T + 29T^{2} \)
31 \( 1 - 2.32T + 31T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + 6.40T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 6.27T + 53T^{2} \)
59 \( 1 + 2.72T + 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 1.50T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 0.945T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−10.13716391300802609309052136648, −9.331916306062282560231081080998, −8.498973145566492954161623228048, −7.992616401197181024289997434457, −6.84096789915199637976782712864, −6.40550248589223874724653859772, −4.40533551715497340336436394663, −3.25023180569680189903146223395, −1.56542435205385803237484305376, 0, 1.56542435205385803237484305376, 3.25023180569680189903146223395, 4.40533551715497340336436394663, 6.40550248589223874724653859772, 6.84096789915199637976782712864, 7.992616401197181024289997434457, 8.498973145566492954161623228048, 9.331916306062282560231081080998, 10.13716391300802609309052136648

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