Properties

Label 4-1200e2-1.1-c1e2-0-40
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 8·11-s − 8·19-s − 12·29-s − 16·31-s − 12·41-s + 14·49-s + 8·59-s − 4·61-s − 16·71-s − 16·79-s + 81-s + 12·89-s + 8·99-s − 36·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 2.41·11-s − 1.83·19-s − 2.22·29-s − 2.87·31-s − 1.87·41-s + 2·49-s + 1.04·59-s − 0.512·61-s − 1.89·71-s − 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 3.58·101-s + 0.383·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−9.442517922138744862066974570191, −9.095569430131749971545586709374, −8.681755263756113727855447818436, −8.421824338907487617689049402103, −7.73362334914342770735122976781, −7.65405360595228567799146346241, −7.10487299799159393557263403488, −6.80311028026961086387750024953, −6.00872006837976967030667424282, −5.58338788664374810301843404421, −5.39975766186818179612635171872, −5.05554635793203841842230107205, −4.09791195080334812116643820526, −4.03478677402135871280048824257, −3.21109854637721406124469186890, −2.73704710677610881646288420532, −2.04555220160031712880904490069, −1.80509395194441917922238338027, 0, 0, 1.80509395194441917922238338027, 2.04555220160031712880904490069, 2.73704710677610881646288420532, 3.21109854637721406124469186890, 4.03478677402135871280048824257, 4.09791195080334812116643820526, 5.05554635793203841842230107205, 5.39975766186818179612635171872, 5.58338788664374810301843404421, 6.00872006837976967030667424282, 6.80311028026961086387750024953, 7.10487299799159393557263403488, 7.65405360595228567799146346241, 7.73362334914342770735122976781, 8.421824338907487617689049402103, 8.681755263756113727855447818436, 9.095569430131749971545586709374, 9.442517922138744862066974570191

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