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Properties

Label 2-432-3.2-c0-0-0
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 13-s + 19-s + 25-s − 2·31-s − 37-s − 2·43-s − 61-s + 67-s − 73-s + 79-s − 91-s − 97-s + 103-s + 2·109-s + ⋯
L(s)  = 1  + 7-s − 13-s + 19-s + 25-s − 2·31-s − 37-s − 2·43-s − 61-s + 67-s − 73-s + 79-s − 91-s − 97-s + 103-s + 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{432} (161, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9011863890\)
\(L(\frac12)\) \(\approx\) \(0.9011863890\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−11.39234239122914693129514816042, −10.54430895002755349161485655192, −9.566302580069203901949205746242, −8.636821193411037435011223717629, −7.65324734815483488908880335481, −6.91161773717674793494246491444, −5.41095967352828496575967212386, −4.77708743694400516348693001663, −3.31102455831743181224971400409, −1.79686130737833177314102255441, 1.79686130737833177314102255441, 3.31102455831743181224971400409, 4.77708743694400516348693001663, 5.41095967352828496575967212386, 6.91161773717674793494246491444, 7.65324734815483488908880335481, 8.636821193411037435011223717629, 9.566302580069203901949205746242, 10.54430895002755349161485655192, 11.39234239122914693129514816042

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