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Properties

Label 2-432-12.11-c3-0-23
Degree $2$
Conductor $432$
Sign $-0.866 - 0.5i$
Analytic cond. $25.4888$
Root an. cond. $5.04864$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32.9i·7-s − 89·13-s + 126. i·19-s + 125·25-s + 155. i·31-s − 433·37-s + 218. i·43-s − 740·49-s − 901·61-s − 434. i·67-s + 271·73-s + 1.31e3i·79-s + 2.92e3i·91-s − 1.85e3·97-s − 2.09e3i·103-s + ⋯
L(s)  = 1  − 1.77i·7-s − 1.89·13-s + 1.52i·19-s + 25-s + 0.903i·31-s − 1.92·37-s + 0.773i·43-s − 2.15·49-s − 1.89·61-s − 0.792i·67-s + 0.434·73-s + 1.86i·79-s + 3.37i·91-s − 1.93·97-s − 1.99i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(25.4888\)
Root analytic conductor: \(5.04864\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 432,\ (\ :3/2),\ -0.866 - 0.5i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 125T^{2} \)
7 \( 1 + 32.9iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 89T + 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 126. iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + 433T + 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 218. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 901T + 2.26e5T^{2} \)
67 \( 1 + 434. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 271T + 3.89e5T^{2} \)
79 \( 1 - 1.31e3iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 + 1.85e3T + 9.12e5T^{2} \)
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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

−10.27664903676889688649347679949, −9.560945575164827817792960325744, −8.176653614018976379689498973269, −7.34297777075260599005533846303, −6.72967014436714130385523910118, −5.19403733282421811535969846355, −4.29648735786399857592366647550, −3.17222073629957670720887843289, −1.48692194546584901920359089368, 0, 2.17654932031609721244871505393, 2.92025749855875316428798863863, 4.79149732185335811587906606616, 5.37670338493954760822069076724, 6.60910255470084192685458328411, 7.55239573072325795768770326307, 8.817232735287914562414151096961, 9.234110190151309942324843667191, 10.28657176388879190146371166068

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