Properties

Label 2-432-3.2-c2-0-9
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  + 13·7-s − 13-s − 11·19-s + 25·25-s + 46·31-s + 47·37-s + 22·43-s + 120·49-s − 121·61-s + 109·67-s − 97·73-s − 131·79-s − 13·91-s + 167·97-s + 37·103-s − 214·109-s + ⋯
L(s)  = 1  + 13/7·7-s − 0.0769·13-s − 0.578·19-s + 25-s + 1.48·31-s + 1.27·37-s + 0.511·43-s + 2.44·49-s − 1.98·61-s + 1.62·67-s − 1.32·73-s − 1.65·79-s − 1/7·91-s + 1.72·97-s + 0.359·103-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

\(L(\frac{3}{2})\) \(\approx\) \(2.120417464\)
\(L(\frac12)\) \(\approx\) \(2.120417464\)
\(L(2)\) 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 \( 1 - 13 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 11 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 46 T + p^{2} T^{2} \)
37 \( 1 - 47 T + p^{2} T^{2} \)
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 \( 1 + 121 T + p^{2} T^{2} \)
67 \( 1 - 109 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 + 97 T + p^{2} T^{2} \)
79 \( 1 + 131 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 167 T + p^{2} T^{2} \)
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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

−11.03026690092992823481817781343, −10.21492171622462054868727733765, −8.933561478699626150595039599215, −8.199196890026437243064339513920, −7.42957089512152402146815923745, −6.17544996740251869284882079390, −4.97610347552061534285370424412, −4.30372489888907407074895129107, −2.55900693370604503446143988447, −1.23038637946345969613229333503, 1.23038637946345969613229333503, 2.55900693370604503446143988447, 4.30372489888907407074895129107, 4.97610347552061534285370424412, 6.17544996740251869284882079390, 7.42957089512152402146815923745, 8.199196890026437243064339513920, 8.933561478699626150595039599215, 10.21492171622462054868727733765, 11.03026690092992823481817781343

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