Properties

Degree $2$
Conductor $47040$
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 + 5-s + 9-s + 2·13-s + 15-s + 6·17-s + 8·19-s + 25-s + 27-s − 6·29-s + 4·31-s + 10·37-s + 2·39-s + 6·41-s + 4·43-s + 45-s + 6·51-s + 6·53-s + 8·57-s − 12·59-s − 10·61-s + 2·65-s + 4·67-s + 12·71-s + 10·73-s + 75-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 1.64·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.840·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.115·75-s + 0.900·79-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: \(0\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.863140298\)
\(L(\frac12)\) \(\approx\) \(4.863140298\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.52450798177665, −14.01603784106283, −13.65397752938481, −13.23682451678266, −12.46430679134715, −12.18312387947942, −11.47494024470777, −10.94625019478611, −10.39101522908668, −9.637890704901185, −9.467671422517541, −9.027400990236111, −8.049249223160365, −7.746330187862586, −7.413336128190400, −6.450185096590153, −6.004978283148494, −5.377776103465982, −4.892758583220991, −3.958095577540699, −3.486798944148466, −2.869205824206551, −2.238878011625364, −1.245738535463357, −0.8837473387527490, 0.8837473387527490, 1.245738535463357, 2.238878011625364, 2.869205824206551, 3.486798944148466, 3.958095577540699, 4.892758583220991, 5.377776103465982, 6.004978283148494, 6.450185096590153, 7.413336128190400, 7.746330187862586, 8.049249223160365, 9.027400990236111, 9.467671422517541, 9.637890704901185, 10.39101522908668, 10.94625019478611, 11.47494024470777, 12.18312387947942, 12.46430679134715, 13.23682451678266, 13.65397752938481, 14.01603784106283, 14.52450798177665

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