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Properties

Label 2-864-24.11-c3-0-12
Degree $2$
Conductor $864$
Sign $-0.273 - 0.961i$
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  + 13.3·5-s − 1.40i·7-s − 17.6i·11-s + 70.2i·13-s + 79.0i·17-s − 107.·19-s + 41.7·23-s + 52.4·25-s − 282.·29-s + 84.6i·31-s − 18.6i·35-s + 292. i·37-s + 108. i·41-s − 29.7·43-s + 206.·47-s + ⋯
L(s)  = 1  + 1.19·5-s − 0.0757i·7-s − 0.484i·11-s + 1.49i·13-s + 1.12i·17-s − 1.29·19-s + 0.378·23-s + 0.419·25-s − 1.81·29-s + 0.490i·31-s − 0.0902i·35-s + 1.29i·37-s + 0.415i·41-s − 0.105·43-s + 0.641·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.273 - 0.961i$
Analytic conductor: \(50.9776\)
Root analytic conductor: \(7.13986\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :3/2),\ -0.273 - 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.720031078\)
\(L(\frac12)\) \(\approx\) \(1.720031078\)
\(L(\frac{5}{2})\) 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 \( 1 - 13.3T + 125T^{2} \)
7 \( 1 + 1.40iT - 343T^{2} \)
11 \( 1 + 17.6iT - 1.33e3T^{2} \)
13 \( 1 - 70.2iT - 2.19e3T^{2} \)
17 \( 1 - 79.0iT - 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 - 41.7T + 1.21e4T^{2} \)
29 \( 1 + 282.T + 2.43e4T^{2} \)
31 \( 1 - 84.6iT - 2.97e4T^{2} \)
37 \( 1 - 292. iT - 5.06e4T^{2} \)
41 \( 1 - 108. iT - 6.89e4T^{2} \)
43 \( 1 + 29.7T + 7.95e4T^{2} \)
47 \( 1 - 206.T + 1.03e5T^{2} \)
53 \( 1 - 207.T + 1.48e5T^{2} \)
59 \( 1 - 618. iT - 2.05e5T^{2} \)
61 \( 1 + 667. iT - 2.26e5T^{2} \)
67 \( 1 + 848.T + 3.00e5T^{2} \)
71 \( 1 + 535.T + 3.57e5T^{2} \)
73 \( 1 - 397.T + 3.89e5T^{2} \)
79 \( 1 + 1.19e3iT - 4.93e5T^{2} \)
83 \( 1 - 636. iT - 5.71e5T^{2} \)
89 \( 1 + 179. iT - 7.04e5T^{2} \)
97 \( 1 - 550.T + 9.12e5T^{2} \)
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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

−10.04738206726968278831300192180, −9.080885921565322200804596735145, −8.653332171902880516783472449708, −7.35821838427186804425197515274, −6.34243801552472165595291662176, −5.91197945460964134756143005488, −4.68831959014938715389613418376, −3.70242860447082255121969485774, −2.24156882105537218013219801175, −1.49807090872205133379425249217, 0.41006816494754457642939353586, 1.92390985432214579880370358471, 2.75584395171756832471870122704, 4.12136051831730638615956903794, 5.41934923471536401721069591998, 5.76925584586701249323586237773, 6.97610901044331600306765707134, 7.74593635982290674468729348089, 8.916992491867009167383161272800, 9.503589423850276977262716034557

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