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Properties

Label 2-432-4.3-c8-0-53
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  + 612.·5-s + 3.29e3i·7-s + 5.14e3i·11-s + 2.62e4·13-s + 1.55e5·17-s − 1.21e5i·19-s − 4.97e5i·23-s − 1.52e4·25-s − 1.00e6·29-s − 1.51e6i·31-s + 2.01e6i·35-s − 2.43e6·37-s + 5.45e6·41-s − 2.74e6i·43-s − 8.83e6i·47-s + ⋯
L(s)  = 1  + 0.980·5-s + 1.37i·7-s + 0.351i·11-s + 0.918·13-s + 1.86·17-s − 0.929i·19-s − 1.77i·23-s − 0.0389·25-s − 1.42·29-s − 1.64i·31-s + 1.34i·35-s − 1.30·37-s + 1.93·41-s − 0.803i·43-s − 1.81i·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

\(L(\frac{9}{2})\) \(\approx\) \(3.057498939\)
\(L(\frac12)\) \(\approx\) \(3.057498939\)
\(L(5)\) 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 \( 1 - 612.T + 3.90e5T^{2} \)
7 \( 1 - 3.29e3iT - 5.76e6T^{2} \)
11 \( 1 - 5.14e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.62e4T + 8.15e8T^{2} \)
17 \( 1 - 1.55e5T + 6.97e9T^{2} \)
19 \( 1 + 1.21e5iT - 1.69e10T^{2} \)
23 \( 1 + 4.97e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.00e6T + 5.00e11T^{2} \)
31 \( 1 + 1.51e6iT - 8.52e11T^{2} \)
37 \( 1 + 2.43e6T + 3.51e12T^{2} \)
41 \( 1 - 5.45e6T + 7.98e12T^{2} \)
43 \( 1 + 2.74e6iT - 1.16e13T^{2} \)
47 \( 1 + 8.83e6iT - 2.38e13T^{2} \)
53 \( 1 - 7.85e6T + 6.22e13T^{2} \)
59 \( 1 + 1.93e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.59e7T + 1.91e14T^{2} \)
67 \( 1 + 1.93e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.01e6iT - 6.45e14T^{2} \)
73 \( 1 + 9.41e6T + 8.06e14T^{2} \)
79 \( 1 - 6.11e7iT - 1.51e15T^{2} \)
83 \( 1 - 2.44e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.51e7T + 3.93e15T^{2} \)
97 \( 1 + 3.81e7T + 7.83e15T^{2} \)
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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

−9.623875476370697412556943259274, −8.971555812019959262812667644988, −8.087747684972705796685075576810, −6.82103858188356085321790384098, −5.69019753445356879783151247653, −5.48054008323605967011590326658, −3.90273939489771117314161582869, −2.58234902664437182160685262639, −1.92993934278012973599489433791, −0.59427089274653308460879715630, 1.11485032412373860218508198867, 1.49910831784488719423380052824, 3.27571685540564585376541350651, 3.89809576241772521587964850891, 5.46799073879750157313813197808, 5.95476205686566875649323862498, 7.26555518820369043949215372428, 7.889091017313233193058389716658, 9.181641557715275286371994094376, 9.989629812967516315693350782829

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