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Properties

Label 2-432-9.2-c2-0-10
Degree $2$
Conductor $432$
Sign $-0.173 + 0.984i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.39 − 1.96i)5-s + (6.39 − 11.0i)7-s + (−14.2 − 8.25i)11-s + (1.39 + 2.42i)13-s − 2.54i·17-s − 21.5·19-s + (2.60 − 1.50i)23-s + (−4.79 + 8.31i)25-s + (13.1 + 7.61i)29-s + (−13.6 − 23.5i)31-s − 50.2i·35-s − 10.4·37-s + (34.5 − 19.9i)41-s + (17.0 − 29.6i)43-s + (58.1 + 33.6i)47-s + ⋯
L(s)  = 1  + (0.679 − 0.392i)5-s + (0.914 − 1.58i)7-s + (−1.29 − 0.750i)11-s + (0.107 + 0.186i)13-s − 0.149i·17-s − 1.13·19-s + (0.113 − 0.0652i)23-s + (−0.191 + 0.332i)25-s + (0.455 + 0.262i)29-s + (−0.438 − 0.759i)31-s − 1.43i·35-s − 0.281·37-s + (0.841 − 0.485i)41-s + (0.397 − 0.688i)43-s + (1.23 + 0.714i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10101 - 1.31214i\)
\(L(\frac12)\) \(\approx\) \(1.10101 - 1.31214i\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-3.39 + 1.96i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-6.39 + 11.0i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (14.2 + 8.25i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.39 - 2.42i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 2.54iT - 289T^{2} \)
19 \( 1 + 21.5T + 361T^{2} \)
23 \( 1 + (-2.60 + 1.50i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.1 - 7.61i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (13.6 + 23.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 10.4T + 1.36e3T^{2} \)
41 \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.0 + 29.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-58.1 - 33.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-5.29 + 3.05i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20.3 - 35.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-54.4 - 94.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 52.8iT - 5.04e3T^{2} \)
73 \( 1 - 68.7T + 5.32e3T^{2} \)
79 \( 1 + (12.7 - 22.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-52.0 - 30.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 7.62iT - 7.92e3T^{2} \)
97 \( 1 + (-50.7 + 87.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.74105298813656743805654758119, −9.995000027831748948367463683419, −8.767407133603791959432596723367, −7.934824800166574123896414903981, −7.09942845457914720134534150038, −5.80077040297752502713329763876, −4.87199954324235508398743635784, −3.83956351015071262929312057889, −2.15243820200788307346026128428, −0.70578592482523940787914997541, 2.01786575154539002977980356957, 2.65930282161934989832252513834, 4.62356614784607801651047292307, 5.50106251309651368607549063664, 6.28665239891537787514119482973, 7.66919616807909272490010407077, 8.463162671325989950609430547227, 9.348398769496681633894465142268, 10.41113667980134381978481665726, 11.02069449839205624914049275644

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