Properties

Label 4-460800-1.1-c1e2-0-68
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 $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 8·11-s − 4·17-s − 16·19-s + 25-s − 4·27-s + 16·33-s − 12·41-s − 8·43-s − 14·49-s + 8·51-s + 32·57-s − 24·59-s + 24·67-s + 4·73-s − 2·75-s + 5·81-s + 8·83-s + 4·89-s − 28·97-s − 24·99-s + 24·107-s + 12·113-s + 26·121-s + 24·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 2.41·11-s − 0.970·17-s − 3.67·19-s + 1/5·25-s − 0.769·27-s + 2.78·33-s − 1.87·41-s − 1.21·43-s − 2·49-s + 1.12·51-s + 4.23·57-s − 3.12·59-s + 2.93·67-s + 0.468·73-s − 0.230·75-s + 5/9·81-s + 0.878·83-s + 0.423·89-s − 2.84·97-s − 2.41·99-s + 2.32·107-s + 1.12·113-s + 2.36·121-s + 2.16·123-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: \(2\)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−7.985644465490789105614221217605, −7.888892617242428206551037061144, −6.84594392053537268258202260884, −6.76366111244607068879792174831, −6.25812475404630879226037514190, −5.86487938441035862983502527784, −5.10335741728591181213017059039, −4.78210199118180492802537017050, −4.59744237493557402533707286052, −3.76484405045814108002996889467, −3.01325599841700254495812174327, −2.15654793577695836540091578775, −1.90410956588849953091073290916, 0, 0, 1.90410956588849953091073290916, 2.15654793577695836540091578775, 3.01325599841700254495812174327, 3.76484405045814108002996889467, 4.59744237493557402533707286052, 4.78210199118180492802537017050, 5.10335741728591181213017059039, 5.86487938441035862983502527784, 6.25812475404630879226037514190, 6.76366111244607068879792174831, 6.84594392053537268258202260884, 7.888892617242428206551037061144, 7.985644465490789105614221217605

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