↑

Properties

Label 8-240e4-1.1-c0e4-0-0
Degree $8$
Conductor $3317760000$
Sign $1$
Analytic cond. $0.000205813$
Root an. cond. $0.346086$
Motivic weight $0$
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  − 16-s − 4·19-s − 4·49-s + 4·61-s + 8·79-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 16-s − 4·19-s − 4·49-s + 4·61-s + 8·79-s − 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.000205813\)
Root analytic conductor: \(0.346086\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2691482008\)
\(L(\frac12)\) \(\approx\) \(0.2691482008\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
good7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$ \( ( 1 - T )^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.152244553984701345408037980079, −8.841082183980231248155593852572, −8.542877462261970093053789337113, −8.332017620077728990333548048250, −8.043005708260171471716271182966, −7.909413203481590687250027947400, −7.80746578592877739633446267261, −7.10643469094892591735622019280, −6.77479150238894859714644007434, −6.64680116357307140519216654282, −6.52446969123786007095903519098, −6.28632739387403651003711591542, −6.10824185492806981195453492813, −5.34247706028749470396480426010, −5.31144419502214469638598438828, −4.88125479089220107901179134265, −4.62319019308263877060332591515, −4.41427234721809658050812687514, −3.82307167587796902320542056164, −3.67167826288751907376634774032, −3.53036680717894158420722420130, −2.53553620062341549250040701672, −2.30617234132287283414117722721, −2.24575443364944395837814610849, −1.51939273448247247661441585868, 1.51939273448247247661441585868, 2.24575443364944395837814610849, 2.30617234132287283414117722721, 2.53553620062341549250040701672, 3.53036680717894158420722420130, 3.67167826288751907376634774032, 3.82307167587796902320542056164, 4.41427234721809658050812687514, 4.62319019308263877060332591515, 4.88125479089220107901179134265, 5.31144419502214469638598438828, 5.34247706028749470396480426010, 6.10824185492806981195453492813, 6.28632739387403651003711591542, 6.52446969123786007095903519098, 6.64680116357307140519216654282, 6.77479150238894859714644007434, 7.10643469094892591735622019280, 7.80746578592877739633446267261, 7.909413203481590687250027947400, 8.043005708260171471716271182966, 8.332017620077728990333548048250, 8.542877462261970093053789337113, 8.841082183980231248155593852572, 9.152244553984701345408037980079

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