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Properties

Label 2-432-36.31-c8-0-0
Degree $2$
Conductor $432$
Sign $-0.905 + 0.424i$
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  + (195. − 339. i)5-s + (−2.55e3 + 1.47e3i)7-s + (3.04e3 − 1.75e3i)11-s + (−1.99e4 + 3.46e4i)13-s + 1.00e4·17-s + 7.08e4i·19-s + (1.69e5 + 9.81e4i)23-s + (1.18e5 + 2.05e5i)25-s + (−9.96e4 − 1.72e5i)29-s + (7.59e4 + 4.38e4i)31-s + 1.15e6i·35-s + 1.15e6·37-s + (2.15e6 − 3.74e6i)41-s + (−2.25e6 + 1.30e6i)43-s + (−3.82e6 + 2.21e6i)47-s + ⋯
L(s)  = 1  + (0.313 − 0.543i)5-s + (−1.06 + 0.614i)7-s + (0.207 − 0.119i)11-s + (−0.700 + 1.21i)13-s + 0.120·17-s + 0.543i·19-s + (0.607 + 0.350i)23-s + (0.303 + 0.525i)25-s + (−0.140 − 0.243i)29-s + (0.0822 + 0.0474i)31-s + 0.770i·35-s + 0.617·37-s + (0.764 − 1.32i)41-s + (−0.658 + 0.380i)43-s + (−0.784 + 0.452i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(175.987\)
Root analytic conductor: \(13.2660\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :4),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.03319067658\)
\(L(\frac12)\) \(\approx\) \(0.03319067658\)
\(L(5)\) 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 \( 1 + (-195. + 339. i)T + (-1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (2.55e3 - 1.47e3i)T + (2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (-3.04e3 + 1.75e3i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + (1.99e4 - 3.46e4i)T + (-4.07e8 - 7.06e8i)T^{2} \)
17 \( 1 - 1.00e4T + 6.97e9T^{2} \)
19 \( 1 - 7.08e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.69e5 - 9.81e4i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + (9.96e4 + 1.72e5i)T + (-2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (-7.59e4 - 4.38e4i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 - 1.15e6T + 3.51e12T^{2} \)
41 \( 1 + (-2.15e6 + 3.74e6i)T + (-3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (2.25e6 - 1.30e6i)T + (5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (3.82e6 - 2.21e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 - 6.50e6T + 6.22e13T^{2} \)
59 \( 1 + (-1.16e7 - 6.71e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (7.38e6 + 1.27e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.66e7 + 9.63e6i)T + (2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 2.02e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.22e7T + 8.06e14T^{2} \)
79 \( 1 + (-5.09e7 + 2.93e7i)T + (7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (7.50e7 - 4.33e7i)T + (1.12e15 - 1.95e15i)T^{2} \)
89 \( 1 - 3.85e7T + 3.93e15T^{2} \)
97 \( 1 + (5.66e6 + 9.81e6i)T + (-3.91e15 + 6.78e15i)T^{2} \)
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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

−10.06651157742898196220435612527, −9.314439545117701115997847532142, −8.873216053007260015207333962345, −7.51510722288937468267697174185, −6.55398974315027686576725724871, −5.72159662869792171132740231078, −4.70726245854176757499652012550, −3.54105958226477783604149685266, −2.43234675810977158617739995406, −1.34529477794781423044431718170, 0.00699715901814842167015633186, 0.954319374143320409164184190913, 2.59225509945759060806876896834, 3.20590579049027221978980921873, 4.45817063812899167469803131588, 5.63571493343988797212763189148, 6.64991460696678549287922620771, 7.24007554516902092678546462589, 8.372237019458484057808320844949, 9.580157265287020750351515026636

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