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Properties

Label 2-432-9.2-c4-0-6
Degree $2$
Conductor $432$
Sign $-0.199 - 0.979i$
Analytic cond. $44.6558$
Root an. cond. $6.68250$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.6 + 6.14i)5-s + (−7.14 + 12.3i)7-s + (90.2 + 52.0i)11-s + (−37.6 − 65.1i)13-s − 341. i·17-s + 706.·19-s + (−516. + 298. i)23-s + (−237. + 410. i)25-s + (−1.12e3 − 651. i)29-s + (514. + 891. i)31-s − 175. i·35-s + 563.·37-s + (−85.8 + 49.5i)41-s + (−448. + 776. i)43-s + (372. + 215. i)47-s + ⋯
L(s)  = 1  + (−0.425 + 0.245i)5-s + (−0.145 + 0.252i)7-s + (0.745 + 0.430i)11-s + (−0.222 − 0.385i)13-s − 1.18i·17-s + 1.95·19-s + (−0.976 + 0.563i)23-s + (−0.379 + 0.657i)25-s + (−1.34 − 0.774i)29-s + (0.535 + 0.927i)31-s − 0.143i·35-s + 0.411·37-s + (−0.0510 + 0.0294i)41-s + (−0.242 + 0.419i)43-s + (0.168 + 0.0974i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.199 - 0.979i$
Analytic conductor: \(44.6558\)
Root analytic conductor: \(6.68250\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :2),\ -0.199 - 0.979i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.314338417\)
\(L(\frac12)\) \(\approx\) \(1.314338417\)
\(L(3)\) 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 \( 1 + (10.6 - 6.14i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (7.14 - 12.3i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-90.2 - 52.0i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (37.6 + 65.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 341. iT - 8.35e4T^{2} \)
19 \( 1 - 706.T + 1.30e5T^{2} \)
23 \( 1 + (516. - 298. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.12e3 + 651. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-514. - 891. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 563.T + 1.87e6T^{2} \)
41 \( 1 + (85.8 - 49.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (448. - 776. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-372. - 215. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 5.27e3iT - 7.89e6T^{2} \)
59 \( 1 + (4.88e3 - 2.81e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (565. - 979. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (676. + 1.17e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.68e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.23e3T + 2.83e7T^{2} \)
79 \( 1 + (3.06e3 - 5.31e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-6.50e3 - 3.75e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 8.72e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.72e3 - 4.71e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−10.90153719685169138195573874872, −9.565122772050478033201846800940, −9.355469520103819598790980682772, −7.72206630923146955202411030167, −7.35723941802333149710164384999, −6.04573411525557511497725317663, −5.05841478669423568512409153115, −3.80156914912538931853287341823, −2.78547820675122921009470697780, −1.20336882203244719861526243869, 0.39573368036304006695195892302, 1.76884486664680720646111087359, 3.44706636732888911052780388837, 4.21998400553588155549141965743, 5.54477792892656618039970392124, 6.52062355429559210062858952884, 7.59223054689261274116701922173, 8.402440063826757166624510320473, 9.424827443174472030613808632793, 10.19065564478101687494297778620

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