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Properties

Label 2-432-4.3-c8-0-13
Degree $2$
Conductor $432$
Sign $0.866 - 0.5i$
Analytic cond. $175.987$
Root an. cond. $13.2660$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 281.·5-s + 23.3i·7-s − 1.84e4i·11-s − 4.47e4·13-s − 9.43e4·17-s − 7.80e4i·19-s + 5.01e5i·23-s − 3.11e5·25-s − 6.37e5·29-s − 3.95e5i·31-s − 6.58e3i·35-s + 1.01e6·37-s + 3.04e6·41-s + 2.55e6i·43-s − 7.20e6i·47-s + ⋯
L(s)  = 1  − 0.450·5-s + 0.00974i·7-s − 1.25i·11-s − 1.56·13-s − 1.12·17-s − 0.598i·19-s + 1.79i·23-s − 0.797·25-s − 0.900·29-s − 0.427i·31-s − 0.00438i·35-s + 0.539·37-s + 1.07·41-s + 0.746i·43-s − 1.47i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(175.987\)
Root analytic conductor: \(13.2660\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :4),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.7598595158\)
\(L(\frac12)\) \(\approx\) \(0.7598595158\)
\(L(5)\) 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 \( 1 + 281.T + 3.90e5T^{2} \)
7 \( 1 - 23.3iT - 5.76e6T^{2} \)
11 \( 1 + 1.84e4iT - 2.14e8T^{2} \)
13 \( 1 + 4.47e4T + 8.15e8T^{2} \)
17 \( 1 + 9.43e4T + 6.97e9T^{2} \)
19 \( 1 + 7.80e4iT - 1.69e10T^{2} \)
23 \( 1 - 5.01e5iT - 7.83e10T^{2} \)
29 \( 1 + 6.37e5T + 5.00e11T^{2} \)
31 \( 1 + 3.95e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.01e6T + 3.51e12T^{2} \)
41 \( 1 - 3.04e6T + 7.98e12T^{2} \)
43 \( 1 - 2.55e6iT - 1.16e13T^{2} \)
47 \( 1 + 7.20e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.51e7T + 6.22e13T^{2} \)
59 \( 1 + 1.76e5iT - 1.46e14T^{2} \)
61 \( 1 + 1.37e7T + 1.91e14T^{2} \)
67 \( 1 + 3.67e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.04e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.27e7T + 8.06e14T^{2} \)
79 \( 1 - 4.53e7iT - 1.51e15T^{2} \)
83 \( 1 - 6.30e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.79e7T + 3.93e15T^{2} \)
97 \( 1 - 7.73e7T + 7.83e15T^{2} \)
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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

−9.702959421636359908229869916921, −9.113984066464192038813212544444, −7.925625453975817470039447103697, −7.31709876732678135771473199220, −6.13423138327412224290229290838, −5.16863591311533312719561468296, −4.10804722664985097985054710049, −3.06326494387686766372454235484, −1.97790411690950737205420658362, −0.48226450369701853598741671435, 0.26312322410360832262044499374, 1.88482538054186747358867199489, 2.67938153478269627958080881669, 4.26047846816463022452864513826, 4.69077514985275635746848467321, 6.07799189589720660231824991955, 7.18195427631815113601796584054, 7.70781913020458304874852248302, 8.913745699697387307267572718748, 9.765007410915727818932630293938

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