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Properties

Label 2-432-9.2-c2-0-6
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  + (−6.39 + 3.69i)5-s + (−3.39 + 5.88i)7-s + (5.29 + 3.05i)11-s + (−8.39 − 14.5i)13-s − 25.1i·17-s + 17.5·19-s + (12.3 − 7.15i)23-s + (14.7 − 25.6i)25-s + (−16.1 − 9.35i)29-s + (−23.3 − 40.5i)31-s − 50.2i·35-s − 49.5·37-s + (34.5 − 19.9i)41-s + (−22.0 + 38.2i)43-s + (28.8 + 16.6i)47-s + ⋯
L(s)  = 1  + (−1.27 + 0.738i)5-s + (−0.485 + 0.841i)7-s + (0.481 + 0.278i)11-s + (−0.646 − 1.11i)13-s − 1.48i·17-s + 0.926·19-s + (0.539 − 0.311i)23-s + (0.591 − 1.02i)25-s + (−0.558 − 0.322i)29-s + (−0.754 − 1.30i)31-s − 1.43i·35-s − 1.34·37-s + (0.841 − 0.485i)41-s + (−0.513 + 0.890i)43-s + (0.612 + 0.353i)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

\(L(\frac{3}{2})\) \(\approx\) \(0.339018 - 0.404026i\)
\(L(\frac12)\) \(\approx\) \(0.339018 - 0.404026i\)
\(L(2)\) 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 \( 1 + (6.39 - 3.69i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.39 - 5.88i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.29 - 3.05i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (8.39 + 14.5i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 25.1iT - 289T^{2} \)
19 \( 1 - 17.5T + 361T^{2} \)
23 \( 1 + (-12.3 + 7.15i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (16.1 + 9.35i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.3 + 40.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 49.5T + 1.36e3T^{2} \)
41 \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (22.0 - 38.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-28.8 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 10.1iT - 2.80e3T^{2} \)
59 \( 1 + (14.2 - 8.25i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (10.6 - 18.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (43.4 + 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 30.2iT - 5.04e3T^{2} \)
73 \( 1 + 48.7T + 5.32e3T^{2} \)
79 \( 1 + (-55.7 + 96.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (85.0 + 49.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 75.5iT - 7.92e3T^{2} \)
97 \( 1 + (-70.2 + 121. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.83981699917439248493041887868, −9.717287997848586580749626169659, −8.967475735193477853285256884002, −7.56815286559497715834819810098, −7.32140899650579445272995559863, −5.95708051853623783458306795656, −4.83341134545110310447171558243, −3.45592004482394501947777154536, −2.68383763551765627182641091925, −0.24118316578607707151304080353, 1.33301649413595707643002438697, 3.55062361428722975594833719076, 4.11230681708829124920011807807, 5.28962584591007486068151929765, 6.81935945301705152760363547873, 7.40711610405656000475923044311, 8.523432115577673431142006307970, 9.220664594104401931046695745740, 10.36973482040562867840803850183, 11.28685456395632564825368792568

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