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Properties

Label 2-432-4.3-c8-0-46
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  + 891.·5-s − 2.85e3i·7-s + 3.64e3i·11-s + 3.27e4·13-s − 1.09e5·17-s + 8.88e3i·19-s + 1.51e3i·23-s + 4.03e5·25-s + 1.11e6·29-s + 8.31e5i·31-s − 2.54e6i·35-s + 9.19e5·37-s + 3.16e6·41-s + 5.51e6i·43-s + 3.07e6i·47-s + ⋯
L(s)  = 1  + 1.42·5-s − 1.19i·7-s + 0.248i·11-s + 1.14·13-s − 1.31·17-s + 0.0681i·19-s + 0.00543i·23-s + 1.03·25-s + 1.58·29-s + 0.900i·31-s − 1.69i·35-s + 0.490·37-s + 1.12·41-s + 1.61i·43-s + 0.629i·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

\(L(\frac{9}{2})\) \(\approx\) \(3.493244176\)
\(L(\frac12)\) \(\approx\) \(3.493244176\)
\(L(5)\) 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 \( 1 - 891.T + 3.90e5T^{2} \)
7 \( 1 + 2.85e3iT - 5.76e6T^{2} \)
11 \( 1 - 3.64e3iT - 2.14e8T^{2} \)
13 \( 1 - 3.27e4T + 8.15e8T^{2} \)
17 \( 1 + 1.09e5T + 6.97e9T^{2} \)
19 \( 1 - 8.88e3iT - 1.69e10T^{2} \)
23 \( 1 - 1.51e3iT - 7.83e10T^{2} \)
29 \( 1 - 1.11e6T + 5.00e11T^{2} \)
31 \( 1 - 8.31e5iT - 8.52e11T^{2} \)
37 \( 1 - 9.19e5T + 3.51e12T^{2} \)
41 \( 1 - 3.16e6T + 7.98e12T^{2} \)
43 \( 1 - 5.51e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.07e6iT - 2.38e13T^{2} \)
53 \( 1 - 7.07e6T + 6.22e13T^{2} \)
59 \( 1 + 1.91e7iT - 1.46e14T^{2} \)
61 \( 1 - 4.78e6T + 1.91e14T^{2} \)
67 \( 1 + 3.07e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.49e6iT - 6.45e14T^{2} \)
73 \( 1 - 3.05e7T + 8.06e14T^{2} \)
79 \( 1 + 3.93e6iT - 1.51e15T^{2} \)
83 \( 1 + 8.57e6iT - 2.25e15T^{2} \)
89 \( 1 + 2.85e7T + 3.93e15T^{2} \)
97 \( 1 + 1.67e7T + 7.83e15T^{2} \)
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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

−9.789897887806221097173070238438, −8.988539414801704180632015988636, −7.953642347785634071994763355392, −6.64493131250365559028362196326, −6.28146573764014547523554304844, −4.96878950245787698161567641615, −4.04931980736399688646976838206, −2.73410536855731999132312902913, −1.60796882189697664016761462686, −0.78731864325179016702465350950, 0.887231231341361859932075493637, 2.09983971626007150917055161826, 2.66322164878510815978524054706, 4.20200625834741270296572175293, 5.52404153204810227414359598510, 5.98667193772693337614135319842, 6.84229346793965237157099675306, 8.518920798114830766768057412205, 8.878122635247956875379289625310, 9.808599872737552211666138206300

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