Properties

Label 4-840e2-1.1-c1e2-0-43
Degree $4$
Conductor $705600$
Sign $1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 9-s + 8·13-s + 4·17-s + 3·25-s − 4·29-s − 12·37-s + 12·41-s + 2·45-s + 49-s + 16·53-s − 20·61-s + 16·65-s + 8·73-s + 81-s + 8·85-s − 28·89-s + 16·97-s − 12·101-s + 4·109-s + 24·113-s + 8·117-s − 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s + 1/3·9-s + 2.21·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 1.97·37-s + 1.87·41-s + 0.298·45-s + 1/7·49-s + 2.19·53-s − 2.56·61-s + 1.98·65-s + 0.936·73-s + 1/9·81-s + 0.867·85-s − 2.96·89-s + 1.62·97-s − 1.19·101-s + 0.383·109-s + 2.25·113-s + 0.739·117-s − 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.029590026\)
\(L(\frac12)\) \(\approx\) \(3.029590026\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.481553612925985512669604062490, −7.81835639585916182571375219056, −7.48284568639593696898416392075, −6.87259354452381471297775882823, −6.51502067396389468143134947504, −5.94033866886860292935603183313, −5.55375053090780051552954501584, −5.45803650924575687262352989502, −4.46195444778293602014826196668, −4.08706078179886277200739673348, −3.43992207551315519554345327677, −3.08927995906889680200761478861, −2.15715671918253819106635126062, −1.54792096213768469866947757162, −0.959800325238339118056269866099, 0.959800325238339118056269866099, 1.54792096213768469866947757162, 2.15715671918253819106635126062, 3.08927995906889680200761478861, 3.43992207551315519554345327677, 4.08706078179886277200739673348, 4.46195444778293602014826196668, 5.45803650924575687262352989502, 5.55375053090780051552954501584, 5.94033866886860292935603183313, 6.51502067396389468143134947504, 6.87259354452381471297775882823, 7.48284568639593696898416392075, 7.81835639585916182571375219056, 8.481553612925985512669604062490

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