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Properties

Label 2-567-63.23-c0-0-0
Degree $2$
Conductor $567$
Sign $0.959 + 0.281i$
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  + 4-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + 16-s + (−1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s − 31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)52-s − 61-s + 64-s − 67-s + ⋯
L(s)  = 1  + 4-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + 16-s + (−1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s − 31-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)52-s − 61-s + 64-s − 67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058870812\)
\(L(\frac12)\) \(\approx\) \(1.058870812\)
\(L(1)\) 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 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.64123496139685372466950323697, −10.44296442957888698253535711773, −9.279922433522336475404770732198, −7.939111179848260716893498977103, −7.46206775078335953647845771706, −6.30366039290600671111492020391, −5.72023158410200712716744824083, −4.02364351838253952612138180346, −3.16885731093290890840972217176, −1.63236657808636826165139741038, 2.02089835968060073777265362640, 2.95341140358247223401633260605, 4.35641852427877435772698579864, 5.74518605003353089268331658647, 6.48568107053539555171484643999, 7.20284394088194346774737559130, 8.485511978040920463016125420904, 9.158662010036836322905447698468, 10.21681859937096691273793211704, 11.16177273363808527727183789979

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