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Properties

Label 2-864-12.11-c3-0-16
Degree $2$
Conductor $864$
Sign $0.707 - 0.707i$
Analytic cond. $50.9776$
Root an. cond. $7.13986$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.95i·5-s − 12.0i·7-s − 22.0·11-s − 57.4·13-s − 97.0i·17-s + 91.8i·19-s + 137.·23-s + 89.5·25-s + 29.4i·29-s + 149. i·31-s + 71.8·35-s + 216.·37-s + 157. i·41-s − 58.4i·43-s − 196.·47-s + ⋯
L(s)  = 1  + 0.532i·5-s − 0.651i·7-s − 0.603·11-s − 1.22·13-s − 1.38i·17-s + 1.10i·19-s + 1.25·23-s + 0.716·25-s + 0.188i·29-s + 0.863i·31-s + 0.346·35-s + 0.962·37-s + 0.599i·41-s − 0.207i·43-s − 0.610·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(50.9776\)
Root analytic conductor: \(7.13986\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.597865065\)
\(L(\frac12)\) \(\approx\) \(1.597865065\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.95iT - 125T^{2} \)
7 \( 1 + 12.0iT - 343T^{2} \)
11 \( 1 + 22.0T + 1.33e3T^{2} \)
13 \( 1 + 57.4T + 2.19e3T^{2} \)
17 \( 1 + 97.0iT - 4.91e3T^{2} \)
19 \( 1 - 91.8iT - 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 - 29.4iT - 2.43e4T^{2} \)
31 \( 1 - 149. iT - 2.97e4T^{2} \)
37 \( 1 - 216.T + 5.06e4T^{2} \)
41 \( 1 - 157. iT - 6.89e4T^{2} \)
43 \( 1 + 58.4iT - 7.95e4T^{2} \)
47 \( 1 + 196.T + 1.03e5T^{2} \)
53 \( 1 + 35.9iT - 1.48e5T^{2} \)
59 \( 1 + 316.T + 2.05e5T^{2} \)
61 \( 1 + 7.87T + 2.26e5T^{2} \)
67 \( 1 - 424. iT - 3.00e5T^{2} \)
71 \( 1 - 350.T + 3.57e5T^{2} \)
73 \( 1 - 678.T + 3.89e5T^{2} \)
79 \( 1 - 1.08e3iT - 4.93e5T^{2} \)
83 \( 1 - 3.90T + 5.71e5T^{2} \)
89 \( 1 - 156. iT - 7.04e5T^{2} \)
97 \( 1 - 1.57e3T + 9.12e5T^{2} \)
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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

−9.966008059365970414155776949962, −9.196506804466302273530356530544, −7.991817882820663155321554152778, −7.26761249560866754667387642679, −6.69008650212509478768199075303, −5.32405128675060109486605141019, −4.65047848377114237385064800686, −3.29350084905611022598980140035, −2.48973561588671924985994414538, −0.874833494702933364919948148327, 0.54013399187183890066888150298, 2.08308891095289542822753199907, 3.01896823090547902289455438867, 4.50913199813616604665044962080, 5.14430030690874005415751974184, 6.11085832989377218873994591713, 7.17344622578332541986616903854, 8.044243657136412170522225881619, 8.900395185295743447154771305444, 9.515695689375916027263349412383

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