Properties

Label 2-432-3.2-c4-0-15
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $44.6558$
Root an. cond. $6.68250$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23·7-s + 191·13-s − 647·19-s + 625·25-s − 194·31-s + 2.59e3·37-s + 3.21e3·43-s − 1.87e3·49-s − 5.23e3·61-s + 8.80e3·67-s + 9.79e3·73-s + 1.23e4·79-s − 4.39e3·91-s + 9.74e3·97-s − 3.43e3·103-s + 2.20e4·109-s + ⋯
L(s)  = 1  − 0.469·7-s + 1.13·13-s − 1.79·19-s + 25-s − 0.201·31-s + 1.89·37-s + 1.73·43-s − 0.779·49-s − 1.40·61-s + 1.96·67-s + 1.83·73-s + 1.98·79-s − 0.530·91-s + 1.03·97-s − 0.323·103-s + 1.85·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(44.6558\)
Root analytic conductor: \(6.68250\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{432} (161, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.864361448\)
\(L(\frac12)\) \(\approx\) \(1.864361448\)
\(L(3)\) 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 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( 1 + 23 T + p^{4} T^{2} \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 - 191 T + p^{4} T^{2} \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( 1 + 647 T + p^{4} T^{2} \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 + 194 T + p^{4} T^{2} \)
37 \( 1 - 2591 T + p^{4} T^{2} \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 - 3214 T + p^{4} T^{2} \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 + 5233 T + p^{4} T^{2} \)
67 \( 1 - 8809 T + p^{4} T^{2} \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( 1 - 9791 T + p^{4} T^{2} \)
79 \( 1 - 12361 T + p^{4} T^{2} \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 - 9743 T + p^{4} T^{2} \)
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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

−10.74266658094348185978317507486, −9.562741325019853304485816935342, −8.747627603800083143161519127330, −7.88280616160315803340710397783, −6.59447469445795556269777830308, −6.02338393396236202075333404311, −4.60950460776271367794115958120, −3.60108894615956467599697521594, −2.30957459986660492721802569466, −0.78151994581548855163194047427, 0.78151994581548855163194047427, 2.30957459986660492721802569466, 3.60108894615956467599697521594, 4.60950460776271367794115958120, 6.02338393396236202075333404311, 6.59447469445795556269777830308, 7.88280616160315803340710397783, 8.747627603800083143161519127330, 9.562741325019853304485816935342, 10.74266658094348185978317507486

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