Properties

Label 8-1200e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $0.128633$
Root an. cond. $0.773872$
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  + 8·61-s − 81-s − 4·121-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 + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  + 8·61-s − 81-s − 4·121-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 + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.128633\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043788109\)
\(L(\frac12)\) \(\approx\) \(1.043788109\)
\(L(1)\) 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 \( 1 \)
3$C_2^2$ \( 1 + T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$ \( ( 1 - T )^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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

−7.09667085380094546767123062037, −6.80058199737320817471580939049, −6.74884801600807357590526138135, −6.65009205907508451557026934974, −6.57738372855317584120674119835, −5.92011309673505568806087375931, −5.72985770508222206291467081173, −5.69567846077450316822158419440, −5.43690937806494213244648617856, −5.12146938351550802448656829725, −5.07473701428459983573087400675, −4.76270548427746030080657825137, −4.35211579153683385784979435733, −4.12131701679073215332867686006, −3.90709732369647258909861137397, −3.81653063637242672878343513336, −3.52784041319034434727517903042, −3.01609200972239756683414937444, −2.98876520402881796355534311222, −2.47595419092273291458371071096, −2.28492571135972876086159921573, −2.07364776671274661445798946880, −1.67850746664169899658497711978, −1.03672170738375410860717161310, −0.874279031667763229495285711211, 0.874279031667763229495285711211, 1.03672170738375410860717161310, 1.67850746664169899658497711978, 2.07364776671274661445798946880, 2.28492571135972876086159921573, 2.47595419092273291458371071096, 2.98876520402881796355534311222, 3.01609200972239756683414937444, 3.52784041319034434727517903042, 3.81653063637242672878343513336, 3.90709732369647258909861137397, 4.12131701679073215332867686006, 4.35211579153683385784979435733, 4.76270548427746030080657825137, 5.07473701428459983573087400675, 5.12146938351550802448656829725, 5.43690937806494213244648617856, 5.69567846077450316822158419440, 5.72985770508222206291467081173, 5.92011309673505568806087375931, 6.57738372855317584120674119835, 6.65009205907508451557026934974, 6.74884801600807357590526138135, 6.80058199737320817471580939049, 7.09667085380094546767123062037

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