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Properties

Label 2-432-9.4-c1-0-1
Degree $2$
Conductor $432$
Sign $0.766 - 0.642i$
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 + 1.73i)7-s + (1.5 + 2.59i)11-s + (−1 + 1.73i)13-s + 3·17-s + 19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (3 + 5.19i)29-s + (−2 + 3.46i)31-s − 4·37-s + (4.5 − 7.79i)41-s + (−0.5 − 0.866i)43-s + (3 + 5.19i)47-s + (1.50 − 2.59i)49-s − 12·53-s + ⋯
L(s)  = 1  + (0.377 + 0.654i)7-s + (0.452 + 0.783i)11-s + (−0.277 + 0.480i)13-s + 0.727·17-s + 0.229·19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (0.557 + 0.964i)29-s + (−0.359 + 0.622i)31-s − 0.657·37-s + (0.702 − 1.21i)41-s + (−0.0762 − 0.132i)43-s + (0.437 + 0.757i)47-s + (0.214 − 0.371i)49-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37041 + 0.498788i\)
\(L(\frac12)\) \(\approx\) \(1.37041 + 0.498788i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25579802640865048040183917587, −10.39754027979041133977890743745, −9.296744252914712871356596893282, −8.717328239856894343614273700497, −7.47832958005682201226848461520, −6.68599060143512744933427306861, −5.40398530316707544838539381833, −4.56105700977336424427672751606, −3.09992969784421783300365466870, −1.66912056630453280328443050680, 1.07858203020415330750621583833, 2.96074728033307913821773537903, 4.11821906729722128344203142504, 5.30129445065071527894209109712, 6.33012584887830489800912173977, 7.49243225827977902162309094611, 8.180894277460319958189576626761, 9.313408029344223610896758252675, 10.19258215256983998838518179889, 11.07866366555740439359771270519

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