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Properties

Label 2-567-63.34-c0-0-0
Degree $2$
Conductor $567$
Sign $-0.766 - 0.642i$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)22-s + (1 + 1.73i)23-s + (−0.5 + 0.866i)25-s + (1 − 1.73i)29-s − 37-s + (0.5 − 0.866i)43-s − 1.99·46-s + (−0.499 − 0.866i)49-s + (−0.499 − 0.866i)50-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)22-s + (1 + 1.73i)23-s + (−0.5 + 0.866i)25-s + (1 − 1.73i)29-s − 37-s + (0.5 − 0.866i)43-s − 1.99·46-s + (−0.499 − 0.866i)49-s + (−0.499 − 0.866i)50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6150028504\)
\(L(\frac12)\) \(\approx\) \(0.6150028504\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - T + T^{2} \)
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 \( 1 - T + T^{2} \)
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

−11.45471948115953840380265067158, −10.04758159011412900474696586721, −9.388438226267083812588762169825, −8.617732192409707026884102272876, −7.63698952781890666863835736023, −7.00206745443248507360652508462, −5.94974794074527406018110746731, −5.18056746607733997958360278028, −3.53142541708543723037775839193, −2.35058894261229666538154699797, 0.866612918483599318875504887179, 2.59302989921513041867618064191, 3.50284371371704468399788061596, 4.91024376047040225448504870602, 6.20362825584017511058747238316, 6.93142908281568460129237422601, 8.290648743424958540160876683876, 8.971442731172250961033400047510, 10.06174511062220923054176795920, 10.61320898013805129958502295299

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