Properties

Label 4-7920e2-1.1-c1e2-0-16
Degree $4$
Conductor $62726400$
Sign $1$
Analytic cond. $3999.48$
Root an. cond. $7.95245$
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  − 2·5-s + 7-s + 2·11-s + 4·13-s − 3·17-s − 19-s + 2·23-s + 3·25-s − 7·29-s − 9·31-s − 2·35-s + 37-s − 4·41-s − 4·43-s − 18·47-s − 9·49-s − 5·53-s − 4·55-s − 6·59-s + 17·61-s − 8·65-s + 4·67-s − 5·71-s − 4·73-s + 2·77-s + 6·79-s + 2·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s + 0.417·23-s + 3/5·25-s − 1.29·29-s − 1.61·31-s − 0.338·35-s + 0.164·37-s − 0.624·41-s − 0.609·43-s − 2.62·47-s − 9/7·49-s − 0.686·53-s − 0.539·55-s − 0.781·59-s + 2.17·61-s − 0.992·65-s + 0.488·67-s − 0.593·71-s − 0.468·73-s + 0.227·77-s + 0.675·79-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(62726400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3999.48\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 62726400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 17 T + 156 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 21 T + 284 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−7.58457854169057470028841080312, −7.49239849729328195465932937746, −6.85696114648667995847584918322, −6.72633103393073266004091860030, −6.24890522141297399451946130037, −6.18707418678238553273257205881, −5.46872103272161474476497025779, −5.13674001962234622844669767450, −4.88533401280455117831328871727, −4.51536497150318530782591768223, −3.92761848324211068144625572683, −3.73449985364262156422424914916, −3.39057917154115975886734368326, −3.17931771598565961749959093966, −2.31005459100045372567236358959, −2.00734894159940210734420942036, −1.41290922155230762443110682275, −1.17787551093557067709920220950, 0, 0, 1.17787551093557067709920220950, 1.41290922155230762443110682275, 2.00734894159940210734420942036, 2.31005459100045372567236358959, 3.17931771598565961749959093966, 3.39057917154115975886734368326, 3.73449985364262156422424914916, 3.92761848324211068144625572683, 4.51536497150318530782591768223, 4.88533401280455117831328871727, 5.13674001962234622844669767450, 5.46872103272161474476497025779, 6.18707418678238553273257205881, 6.24890522141297399451946130037, 6.72633103393073266004091860030, 6.85696114648667995847584918322, 7.49239849729328195465932937746, 7.58457854169057470028841080312

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