Properties

Label 2-864-4.3-c2-0-9
Degree $2$
Conductor $864$
Sign $0.707 - 0.707i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.13·5-s − 4.02i·7-s + 4.30i·11-s − 18.4·13-s + 23.5·17-s − 21.7i·19-s + 30.7i·23-s + 1.34·25-s + 12.6·29-s + 24.5i·31-s + 20.6i·35-s + 18.2·37-s + 38.0·41-s + 34.9i·43-s − 29.6i·47-s + ⋯
L(s)  = 1  − 1.02·5-s − 0.575i·7-s + 0.391i·11-s − 1.42·13-s + 1.38·17-s − 1.14i·19-s + 1.33i·23-s + 0.0536·25-s + 0.437·29-s + 0.791i·31-s + 0.590i·35-s + 0.492·37-s + 0.929·41-s + 0.813i·43-s − 0.630i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.156086435\)
\(L(\frac12)\) \(\approx\) \(1.156086435\)
\(L(2)\) 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 \( 1 + 5.13T + 25T^{2} \)
7 \( 1 + 4.02iT - 49T^{2} \)
11 \( 1 - 4.30iT - 121T^{2} \)
13 \( 1 + 18.4T + 169T^{2} \)
17 \( 1 - 23.5T + 289T^{2} \)
19 \( 1 + 21.7iT - 361T^{2} \)
23 \( 1 - 30.7iT - 529T^{2} \)
29 \( 1 - 12.6T + 841T^{2} \)
31 \( 1 - 24.5iT - 961T^{2} \)
37 \( 1 - 18.2T + 1.36e3T^{2} \)
41 \( 1 - 38.0T + 1.68e3T^{2} \)
43 \( 1 - 34.9iT - 1.84e3T^{2} \)
47 \( 1 + 29.6iT - 2.20e3T^{2} \)
53 \( 1 - 39.3T + 2.80e3T^{2} \)
59 \( 1 - 65.3iT - 3.48e3T^{2} \)
61 \( 1 + 29.8T + 3.72e3T^{2} \)
67 \( 1 - 11.8iT - 4.48e3T^{2} \)
71 \( 1 - 140. iT - 5.04e3T^{2} \)
73 \( 1 - 119.T + 5.32e3T^{2} \)
79 \( 1 + 9.18iT - 6.24e3T^{2} \)
83 \( 1 - 113. iT - 6.88e3T^{2} \)
89 \( 1 - 7.88T + 7.92e3T^{2} \)
97 \( 1 - 55.5T + 9.40e3T^{2} \)
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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

−9.997610269573003338276545529663, −9.389725310748062164578624826690, −8.162946747927589138215391863273, −7.41455004308815967756861002099, −7.04849198399246590140629038609, −5.52818237513769004066313778763, −4.64293969776259587230547441222, −3.74369430118637937663867707288, −2.65853188130862209303165347011, −0.922254457369725660020056139119, 0.50883303364382608112728411481, 2.32465771174988980256254098090, 3.42332582492648342278803055291, 4.42440376470844223053004505403, 5.43824642711047093483108303014, 6.35886962063375280997929912651, 7.70421198423827178803559074306, 7.86398681236203862030943953239, 8.986195124270759377742121339299, 9.901339445783881738409123761903

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