Properties

Degree $2$
Conductor $47040$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s + 4·19-s + 25-s + 27-s + 10·29-s − 4·33-s − 6·37-s − 2·39-s + 6·41-s − 4·43-s + 45-s − 8·47-s − 2·51-s − 6·53-s − 4·55-s + 4·57-s + 4·59-s − 10·61-s − 2·65-s + 4·67-s + 16·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.696·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47040\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{47040} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.78734516259047, −14.20711228684720, −13.85325472025230, −13.42628615561398, −12.83786669984687, −12.36216765948146, −11.91865433325005, −11.01613382375031, −10.74659204599651, −9.977796570921482, −9.728658086239067, −9.184105396471856, −8.341773160897095, −8.146552847462829, −7.504797689480701, −6.810878778725961, −6.453128390145013, −5.518602836205579, −5.056254718862431, −4.627043154571391, −3.739581537467689, −2.957918373766806, −2.631625213820202, −1.901051695976347, −1.056905067803165, 0, 1.056905067803165, 1.901051695976347, 2.631625213820202, 2.957918373766806, 3.739581537467689, 4.627043154571391, 5.056254718862431, 5.518602836205579, 6.453128390145013, 6.810878778725961, 7.504797689480701, 8.146552847462829, 8.341773160897095, 9.184105396471856, 9.728658086239067, 9.977796570921482, 10.74659204599651, 11.01613382375031, 11.91865433325005, 12.36216765948146, 12.83786669984687, 13.42628615561398, 13.85325472025230, 14.20711228684720, 14.78734516259047

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