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Properties

Label 2-567-63.13-c0-0-0
Degree $2$
Conductor $567$
Sign $0.642 + 0.766i$
Analytic cond. $0.282969$
Root an. cond. $0.531949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (0.5 + 0.866i)7-s + 1.73·8-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)22-s + (−0.5 − 0.866i)25-s − 2·28-s + 37-s + (−0.5 − 0.866i)43-s − 3.46·44-s + (−0.499 + 0.866i)49-s + (−0.866 + 1.5i)50-s + ⋯
L(s)  = 1  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (0.5 + 0.866i)7-s + 1.73·8-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)22-s + (−0.5 − 0.866i)25-s − 2·28-s + 37-s + (−0.5 − 0.866i)43-s − 3.46·44-s + (−0.499 + 0.866i)49-s + (−0.866 + 1.5i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5977147800\)
\(L(\frac12)\) \(\approx\) \(0.5977147800\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.89214514367268513131693933435, −9.811412262285066525937874266798, −9.438557081258007918606429047084, −8.501579990467090465047751705009, −7.70792771142513918507748388220, −6.40570697609618678615671053531, −4.87070708722958744869486818323, −3.89043779693755897932940555218, −2.48775217801204080396921726262, −1.65885049347120618410655102747, 1.11999114332158697137500474025, 3.59818317954485894339883335564, 4.88457328426438198487730737810, 5.96802682080223111941735999602, 6.63770565603998512326944184495, 7.64659259947741919184880825894, 8.234661819459637973208029249727, 9.103201546277372344072320035939, 9.849801843384793769381485832274, 10.94347761971671174317014119425

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