Properties

Degree $4$
Conductor $72000$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 9-s − 8·11-s − 8·19-s + 25-s − 4·29-s − 16·31-s − 12·41-s + 45-s − 14·49-s − 8·55-s + 24·59-s + 28·61-s + 16·71-s − 16·79-s + 81-s + 20·89-s − 8·95-s − 8·99-s + 12·101-s − 36·109-s + 26·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.447·5-s + 1/3·9-s − 2.41·11-s − 1.83·19-s + 1/5·25-s − 0.742·29-s − 2.87·31-s − 1.87·41-s + 0.149·45-s − 2·49-s − 1.07·55-s + 3.12·59-s + 3.58·61-s + 1.89·71-s − 1.80·79-s + 1/9·81-s + 2.11·89-s − 0.820·95-s − 0.804·99-s + 1.19·101-s − 3.44·109-s + 2.36·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(72000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{72000} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 72000,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.749205121396688870647577836420, −9.042960162718976648409639393560, −8.323057177226078640880314002796, −8.280524075884956092273610432842, −7.55727707153378484774141976097, −6.89270810769313104844169224865, −6.65963768785974737826480192145, −5.59585158632682227968586342616, −5.36683559095823513937522588289, −4.95639818252911756604562729940, −3.95303834850677407956821293662, −3.42683606015106566673715003849, −2.25002432004047770108316173263, −2.07933314479152275364533906758, 0, 2.07933314479152275364533906758, 2.25002432004047770108316173263, 3.42683606015106566673715003849, 3.95303834850677407956821293662, 4.95639818252911756604562729940, 5.36683559095823513937522588289, 5.59585158632682227968586342616, 6.65963768785974737826480192145, 6.89270810769313104844169224865, 7.55727707153378484774141976097, 8.280524075884956092273610432842, 8.323057177226078640880314002796, 9.042960162718976648409639393560, 9.749205121396688870647577836420

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