Properties

Label 4-600e2-1.1-c1e2-0-38
Degree $4$
Conductor $360000$
Sign $-1$
Analytic cond. $22.9539$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 8·17-s − 12·29-s − 16·37-s − 20·41-s + 2·49-s − 24·53-s + 4·61-s − 16·73-s + 81-s − 20·89-s + 16·97-s − 4·101-s − 4·109-s + 24·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + 167-s − 26·169-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.94·17-s − 2.22·29-s − 2.63·37-s − 3.12·41-s + 2/7·49-s − 3.29·53-s + 0.512·61-s − 1.87·73-s + 1/9·81-s − 2.11·89-s + 1.62·97-s − 0.398·101-s − 0.383·109-s + 2.25·113-s − 0.545·121-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 + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(22.9539\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 360000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−8.663418930560570598737303073543, −7.985267473843071116990190639373, −7.47968045795112434862108262798, −7.22065862859774326573619751365, −6.68900479109315350086651847504, −6.10068275868894969658774395739, −5.49518474552018854569919656545, −5.23638735795602492145985925787, −4.70569070648500642379588704382, −3.87192873599819892504395472144, −3.31973209294614236477974788080, −3.16765979420003827972319820965, −1.72530323216821322522551991647, −1.63474618558680900572990514175, 0, 1.63474618558680900572990514175, 1.72530323216821322522551991647, 3.16765979420003827972319820965, 3.31973209294614236477974788080, 3.87192873599819892504395472144, 4.70569070648500642379588704382, 5.23638735795602492145985925787, 5.49518474552018854569919656545, 6.10068275868894969658774395739, 6.68900479109315350086651847504, 7.22065862859774326573619751365, 7.47968045795112434862108262798, 7.985267473843071116990190639373, 8.663418930560570598737303073543

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