Properties

Label 4-460800-1.1-c1e2-0-59
Degree $4$
Conductor $460800$
Sign $-1$
Analytic cond. $29.3810$
Root an. cond. $2.32818$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 8·11-s + 4·17-s − 8·19-s + 25-s + 4·27-s − 16·33-s + 4·41-s + 8·43-s + 2·49-s + 8·51-s − 16·57-s − 24·59-s − 24·67-s + 20·73-s + 2·75-s + 5·81-s − 8·83-s − 12·89-s − 28·97-s − 24·99-s + 24·107-s − 12·113-s + 26·121-s + 8·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 2.41·11-s + 0.970·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s − 2.78·33-s + 0.624·41-s + 1.21·43-s + 2/7·49-s + 1.12·51-s − 2.11·57-s − 3.12·59-s − 2.93·67-s + 2.34·73-s + 0.230·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 2.84·97-s − 2.41·99-s + 2.32·107-s − 1.12·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(460800\)    =    \(2^{11} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(29.3810\)
Root analytic conductor: \(2.32818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 460800,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.234396477105587706249662636004, −7.75866687525969370360077670275, −7.73435767676089003755122228124, −7.26992741343389521999119886505, −6.39854886739801755961916412419, −6.14269520510095343624413187280, −5.33556450763905259522444606662, −5.11098268655570501466709405394, −4.23445324460423189498619158076, −4.10961287212778855632833672573, −2.94053502252576914533622368524, −2.91524621673910645918598375111, −2.28860875976801017534156160572, −1.47970352314852327908767006362, 0, 1.47970352314852327908767006362, 2.28860875976801017534156160572, 2.91524621673910645918598375111, 2.94053502252576914533622368524, 4.10961287212778855632833672573, 4.23445324460423189498619158076, 5.11098268655570501466709405394, 5.33556450763905259522444606662, 6.14269520510095343624413187280, 6.39854886739801755961916412419, 7.26992741343389521999119886505, 7.73435767676089003755122228124, 7.75866687525969370360077670275, 8.234396477105587706249662636004

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