Properties

Degree $2$
Conductor $58800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 4·11-s + 6·13-s + 2·17-s − 4·19-s + 8·23-s + 27-s − 2·29-s + 4·33-s + 10·37-s + 6·39-s + 6·41-s − 4·43-s + 2·51-s − 6·53-s − 4·57-s + 4·59-s − 6·61-s + 4·67-s + 8·69-s − 8·71-s + 10·73-s + 81-s + 4·83-s − 2·87-s + 6·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.960·39-s + 0.937·41-s − 0.609·43-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.488·67-s + 0.963·69-s − 0.949·71-s + 1.17·73-s + 1/9·81-s + 0.439·83-s − 0.214·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{58800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.699662729\)
\(L(\frac12)\) \(\approx\) \(4.699662729\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.31508379381776, −13.89526827647895, −13.29894000538157, −12.86758428338016, −12.53524797492079, −11.63141450063544, −11.28594800364901, −10.85506754917834, −10.27503806206922, −9.481668178293803, −9.138201192684047, −8.771463859551408, −8.122284022329673, −7.705188270599338, −6.875471343397471, −6.476897959143961, −6.011342564916520, −5.301987299563936, −4.427072856407022, −4.069563392223029, −3.403908307413966, −2.938893464297023, −2.020265449799034, −1.300367912137640, −0.8078845242002148, 0.8078845242002148, 1.300367912137640, 2.020265449799034, 2.938893464297023, 3.403908307413966, 4.069563392223029, 4.427072856407022, 5.301987299563936, 6.011342564916520, 6.476897959143961, 6.875471343397471, 7.705188270599338, 8.122284022329673, 8.771463859551408, 9.138201192684047, 9.481668178293803, 10.27503806206922, 10.85506754917834, 11.28594800364901, 11.63141450063544, 12.53524797492079, 12.86758428338016, 13.29894000538157, 13.89526827647895, 14.31508379381776

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