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Properties

Label 2-432-108.47-c1-0-6
Degree $2$
Conductor $432$
Sign $0.902 + 0.430i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 − 0.517i)3-s + (−2.52 + 0.445i)5-s + (−1.40 + 1.67i)7-s + (2.46 + 1.71i)9-s + (0.751 − 4.26i)11-s + (5.33 + 1.94i)13-s + (4.41 + 0.572i)15-s + (5.53 − 3.19i)17-s + (−2.85 − 1.64i)19-s + (3.19 − 2.04i)21-s + (3.31 − 2.78i)23-s + (1.49 − 0.545i)25-s + (−3.18 − 4.10i)27-s + (0.131 + 0.362i)29-s + (4.37 + 5.21i)31-s + ⋯
L(s)  = 1  + (−0.954 − 0.298i)3-s + (−1.13 + 0.199i)5-s + (−0.532 + 0.634i)7-s + (0.821 + 0.570i)9-s + (0.226 − 1.28i)11-s + (1.47 + 0.538i)13-s + (1.13 + 0.147i)15-s + (1.34 − 0.775i)17-s + (−0.654 − 0.378i)19-s + (0.697 − 0.446i)21-s + (0.691 − 0.580i)23-s + (0.299 − 0.109i)25-s + (−0.613 − 0.789i)27-s + (0.0245 + 0.0673i)29-s + (0.785 + 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797336 - 0.180555i\)
\(L(\frac12)\) \(\approx\) \(0.797336 - 0.180555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.517i)T \)
good5 \( 1 + (2.52 - 0.445i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.40 - 1.67i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.751 + 4.26i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-5.33 - 1.94i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-5.53 + 3.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.85 + 1.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.31 + 2.78i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.131 - 0.362i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.37 - 5.21i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (3.81 + 6.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.138 - 0.380i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-9.35 - 1.64i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.74 - 5.65i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 7.00iT - 53T^{2} \)
59 \( 1 + (-0.296 - 1.68i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.66 - 7.26i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.683 - 1.87i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.85 + 3.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.37 + 9.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.34 + 6.42i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.31 + 0.479i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (6.85 + 3.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.09 + 11.8i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.16251904026956865931938140697, −10.62140194360137092838768359505, −9.136506624514742208328904845468, −8.343361230165173969293246690681, −7.24905554436115614659315384615, −6.31784535665937951297722848930, −5.57437271768196102818652561101, −4.16535746599240570739616570629, −3.09788604456813440300494730743, −0.827928177405748330101367976980, 1.04183750810283082487952910979, 3.70912912742925593465997754713, 4.12751058123559884604707513292, 5.49362610646506141741717275176, 6.52884780216975614100176942864, 7.45478323965504074151902008730, 8.345410868536386956455136482464, 9.718435312397478830613291896951, 10.40375260899572116023356361280, 11.18297649512431139997103689642

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