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Properties

Label 2-864-24.11-c3-0-45
Degree $2$
Conductor $864$
Sign $-0.925 + 0.379i$
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  + 9.32·5-s − 5.38i·7-s − 53.5i·11-s − 57.0i·13-s + 38.0i·17-s − 75.4·19-s − 98.3·23-s − 38.0·25-s − 11.9·29-s − 94.3i·31-s − 50.2i·35-s + 423. i·37-s + 158. i·41-s − 403.·43-s − 119.·47-s + ⋯
L(s)  = 1  + 0.833·5-s − 0.290i·7-s − 1.46i·11-s − 1.21i·13-s + 0.542i·17-s − 0.910·19-s − 0.891·23-s − 0.304·25-s − 0.0766·29-s − 0.546i·31-s − 0.242i·35-s + 1.88i·37-s + 0.605i·41-s − 1.42·43-s − 0.371·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.925 + 0.379i$
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: \((2,\ 864,\ (\ :3/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9926695903\)
\(L(\frac12)\) \(\approx\) \(0.9926695903\)
\(L(\frac{5}{2})\) 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 \( 1 - 9.32T + 125T^{2} \)
7 \( 1 + 5.38iT - 343T^{2} \)
11 \( 1 + 53.5iT - 1.33e3T^{2} \)
13 \( 1 + 57.0iT - 2.19e3T^{2} \)
17 \( 1 - 38.0iT - 4.91e3T^{2} \)
19 \( 1 + 75.4T + 6.85e3T^{2} \)
23 \( 1 + 98.3T + 1.21e4T^{2} \)
29 \( 1 + 11.9T + 2.43e4T^{2} \)
31 \( 1 + 94.3iT - 2.97e4T^{2} \)
37 \( 1 - 423. iT - 5.06e4T^{2} \)
41 \( 1 - 158. iT - 6.89e4T^{2} \)
43 \( 1 + 403.T + 7.95e4T^{2} \)
47 \( 1 + 119.T + 1.03e5T^{2} \)
53 \( 1 + 425.T + 1.48e5T^{2} \)
59 \( 1 + 616. iT - 2.05e5T^{2} \)
61 \( 1 - 58.7iT - 2.26e5T^{2} \)
67 \( 1 - 346.T + 3.00e5T^{2} \)
71 \( 1 + 844.T + 3.57e5T^{2} \)
73 \( 1 - 918.T + 3.89e5T^{2} \)
79 \( 1 + 490. iT - 4.93e5T^{2} \)
83 \( 1 - 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 + 549. iT - 7.04e5T^{2} \)
97 \( 1 - 301.T + 9.12e5T^{2} \)
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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

−9.565885023485745561694883739939, −8.284284703014965543678273603081, −8.098109975635497025904158070864, −6.50677667918095309721646491244, −6.00981424429116678439069808188, −5.13121354925416041329719272209, −3.80964278811472373826547822553, −2.85230178673685043655399015013, −1.55608271638176584355846509731, −0.23398794948469535739348744390, 1.78916693503808233969193479876, 2.28621127048852134666332814269, 3.96237220565833155802245178762, 4.81294278487265984703101531266, 5.83386422509048421568693205543, 6.71771193024684950669500650031, 7.43562258201214708564834310503, 8.656696481792406937673823315825, 9.427348805042005334028397433160, 9.952580992826002917993296692772

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