Properties

Label 2-432-1.1-c1-0-2
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
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 + 5·13-s + 7·19-s − 5·25-s + 4·31-s + 11·37-s − 8·43-s − 6·49-s − 61-s − 5·67-s − 7·73-s − 17·79-s + 5·91-s − 19·97-s + 13·103-s + 2·109-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.38·13-s + 1.60·19-s − 25-s + 0.718·31-s + 1.80·37-s − 1.21·43-s − 6/7·49-s − 0.128·61-s − 0.610·67-s − 0.819·73-s − 1.91·79-s + 0.524·91-s − 1.92·97-s + 1.28·103-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{432} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.529954037\)
\(L(\frac12)\) \(\approx\) \(1.529954037\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−11.38842166918163472638466197964, −10.22208553622788501090350672882, −9.387437094399487619618753359356, −8.341275212548067016845372469550, −7.60583694658959035837478078570, −6.37318226789872439689553131647, −5.48198608682619476325367860976, −4.24977516365535388494352784618, −3.07984083787626508123182523604, −1.36018793518762280095521620239, 1.36018793518762280095521620239, 3.07984083787626508123182523604, 4.24977516365535388494352784618, 5.48198608682619476325367860976, 6.37318226789872439689553131647, 7.60583694658959035837478078570, 8.341275212548067016845372469550, 9.387437094399487619618753359356, 10.22208553622788501090350672882, 11.38842166918163472638466197964

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