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Properties

Label 2-864-72.43-c0-0-0
Degree $2$
Conductor $864$
Sign $0.984 + 0.173i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
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)11-s + 17-s + 19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s − 73-s + (−1 + 1.73i)83-s − 2·89-s + (0.5 − 0.866i)97-s − 107-s + (1 + 1.73i)113-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)11-s + 17-s + 19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s − 73-s + (−1 + 1.73i)83-s − 2·89-s + (0.5 − 0.866i)97-s − 107-s + (1 + 1.73i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038066106\)
\(L(\frac12)\) \(\approx\) \(1.038066106\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
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 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−10.24761749052587498308898050327, −9.541956632536969075482586615462, −8.678611268188822674490573623036, −7.81635340200700342399386201685, −6.97074754312231379452628079929, −5.89213556522097522141364016355, −5.20539249915299923955973645450, −3.84099479174598836544452760295, −3.03524011125714518099321837348, −1.36561928964362376885820479563, 1.53505841668695571911025130471, 2.96620007931651497660331779306, 4.08208851116791577453792807182, 5.08098940866375522330131802136, 6.05116809216466434119450787508, 7.05295618993596042553913882160, 7.78385540983840046143447252941, 8.722677251704992950218099115962, 9.791858392076516277273595432210, 10.08299975763668811066290364970

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