Properties

Label 2-432-3.2-c2-0-6
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·7-s + 23·13-s + 37·19-s + 25·25-s + 46·31-s − 73·37-s + 22·43-s + 72·49-s + 47·61-s + 13·67-s + 143·73-s − 11·79-s − 253·91-s − 169·97-s + 157·103-s − 214·109-s + ⋯
L(s)  = 1  − 1.57·7-s + 1.76·13-s + 1.94·19-s + 25-s + 1.48·31-s − 1.97·37-s + 0.511·43-s + 1.46·49-s + 0.770·61-s + 0.194·67-s + 1.95·73-s − 0.139·79-s − 2.78·91-s − 1.74·97-s + 1.52·103-s − 1.96·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{432} (161, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.565956920\)
\(L(\frac12)\) \(\approx\) \(1.565956920\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 11 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - 23 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 37 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 46 T + p^{2} T^{2} \)
37 \( 1 + 73 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 47 T + p^{2} T^{2} \)
67 \( 1 - 13 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 143 T + p^{2} T^{2} \)
79 \( 1 + 11 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 169 T + p^{2} T^{2} \)
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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

−10.86076580905130567718551924642, −9.989623032380754154401643151522, −9.203468854018199565723952911291, −8.331287839344309215851422522644, −7.01891639391506344634181018241, −6.32241251273778247419460637694, −5.32303470874439342476960852367, −3.71899369818720426939136696502, −3.01597961007327540116277505066, −0.978565515671405865293034079298, 0.978565515671405865293034079298, 3.01597961007327540116277505066, 3.71899369818720426939136696502, 5.32303470874439342476960852367, 6.32241251273778247419460637694, 7.01891639391506344634181018241, 8.331287839344309215851422522644, 9.203468854018199565723952911291, 9.989623032380754154401643151522, 10.86076580905130567718551924642

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