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 + 2·11-s + 2·13-s + 15-s + 4·17-s − 8·23-s + 25-s + 27-s − 2·31-s + 2·33-s − 8·37-s + 2·39-s + 2·41-s − 2·43-s + 45-s + 10·47-s + 4·51-s + 2·53-s + 2·55-s − 4·59-s − 10·61-s + 2·65-s + 2·67-s − 8·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.970·17-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.359·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s + 0.312·41-s − 0.304·43-s + 0.149·45-s + 1.45·47-s + 0.560·51-s + 0.274·53-s + 0.269·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 0.244·67-s − 0.963·69-s + 1.42·71-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.65559040024807, −14.22844963739128, −13.93819919481517, −13.49237392952227, −12.80412792694254, −12.16891102087576, −12.03240696711887, −11.15716658625926, −10.62701761465415, −10.06643110862067, −9.645196945967161, −9.118824811634362, −8.464486461750580, −8.175589957979324, −7.354640480989073, −6.991156364343925, −6.156905994238049, −5.772141030745821, −5.199787748448493, −4.155579927765294, −3.963560442407639, −3.152795743405563, −2.542213361203917, −1.643775129261250, −1.290447694401556, 0, 1.290447694401556, 1.643775129261250, 2.542213361203917, 3.152795743405563, 3.963560442407639, 4.155579927765294, 5.199787748448493, 5.772141030745821, 6.156905994238049, 6.991156364343925, 7.354640480989073, 8.175589957979324, 8.464486461750580, 9.118824811634362, 9.645196945967161, 10.06643110862067, 10.62701761465415, 11.15716658625926, 12.03240696711887, 12.16891102087576, 12.80412792694254, 13.49237392952227, 13.93819919481517, 14.22844963739128, 14.65559040024807

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