Properties

Label 2-432-1.1-c7-0-6
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $134.950$
Root an. cond. $11.6168$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 92.4·5-s − 1.12e3·7-s − 1.11e3·11-s − 9.93e3·13-s + 3.15e4·17-s − 1.90e4·19-s − 4.77e4·23-s − 6.95e4·25-s − 2.52e5·29-s − 4.71e4·31-s + 1.03e5·35-s − 6.27e4·37-s + 5.38e5·41-s − 1.90e5·43-s − 2.91e5·47-s + 4.33e5·49-s + 2.17e5·53-s + 1.03e5·55-s − 2.01e6·59-s − 1.05e6·61-s + 9.18e5·65-s + 1.46e6·67-s − 1.77e6·71-s + 5.31e6·73-s + 1.24e6·77-s + 1.15e6·79-s − 1.53e6·83-s + ⋯
L(s)  = 1  − 0.330·5-s − 1.23·7-s − 0.252·11-s − 1.25·13-s + 1.55·17-s − 0.636·19-s − 0.818·23-s − 0.890·25-s − 1.92·29-s − 0.283·31-s + 0.408·35-s − 0.203·37-s + 1.21·41-s − 0.365·43-s − 0.410·47-s + 0.526·49-s + 0.200·53-s + 0.0834·55-s − 1.27·59-s − 0.593·61-s + 0.414·65-s + 0.595·67-s − 0.588·71-s + 1.59·73-s + 0.311·77-s + 0.262·79-s − 0.295·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(134.950\)
Root analytic conductor: \(11.6168\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6113367747\)
\(L(\frac12)\) \(\approx\) \(0.6113367747\)
\(L(\frac{9}{2})\) 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 \( 1 + 92.4T + 7.81e4T^{2} \)
7 \( 1 + 1.12e3T + 8.23e5T^{2} \)
11 \( 1 + 1.11e3T + 1.94e7T^{2} \)
13 \( 1 + 9.93e3T + 6.27e7T^{2} \)
17 \( 1 - 3.15e4T + 4.10e8T^{2} \)
19 \( 1 + 1.90e4T + 8.93e8T^{2} \)
23 \( 1 + 4.77e4T + 3.40e9T^{2} \)
29 \( 1 + 2.52e5T + 1.72e10T^{2} \)
31 \( 1 + 4.71e4T + 2.75e10T^{2} \)
37 \( 1 + 6.27e4T + 9.49e10T^{2} \)
41 \( 1 - 5.38e5T + 1.94e11T^{2} \)
43 \( 1 + 1.90e5T + 2.71e11T^{2} \)
47 \( 1 + 2.91e5T + 5.06e11T^{2} \)
53 \( 1 - 2.17e5T + 1.17e12T^{2} \)
59 \( 1 + 2.01e6T + 2.48e12T^{2} \)
61 \( 1 + 1.05e6T + 3.14e12T^{2} \)
67 \( 1 - 1.46e6T + 6.06e12T^{2} \)
71 \( 1 + 1.77e6T + 9.09e12T^{2} \)
73 \( 1 - 5.31e6T + 1.10e13T^{2} \)
79 \( 1 - 1.15e6T + 1.92e13T^{2} \)
83 \( 1 + 1.53e6T + 2.71e13T^{2} \)
89 \( 1 - 7.43e6T + 4.42e13T^{2} \)
97 \( 1 - 1.94e6T + 8.07e13T^{2} \)
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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

−9.794579984979057925721949698082, −9.407645518156349050130896497179, −7.933999233113829538406417368624, −7.37191168574205809563219653951, −6.20314184171773519077929062261, −5.36218189603456539898778022066, −4.01089532889130371579139370678, −3.16085054699602505331867057873, −1.99275734558051534311559880620, −0.33414357540699778441641663664, 0.33414357540699778441641663664, 1.99275734558051534311559880620, 3.16085054699602505331867057873, 4.01089532889130371579139370678, 5.36218189603456539898778022066, 6.20314184171773519077929062261, 7.37191168574205809563219653951, 7.933999233113829538406417368624, 9.407645518156349050130896497179, 9.794579984979057925721949698082

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