Properties

Label 8-3920e4-1.1-c0e4-0-3
Degree $8$
Conductor $2.361\times 10^{14}$
Sign $1$
Analytic cond. $14.6478$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 2·25-s + 4·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4·225-s + 227-s + ⋯
L(s)  = 1  + 2·9-s − 2·25-s + 4·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4·225-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(14.6478\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.013339085\)
\(L(\frac12)\) \(\approx\) \(2.013339085\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2$ \( ( 1 - T + T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.15302193392129514545646118067, −6.09975052674398827965950187913, −5.81441181110851486691691011909, −5.51702390320995492468018590750, −5.29037926303853323696174108194, −5.28886286155025898388959890409, −4.84211575406691721862244066989, −4.68206754014833178583060037693, −4.66629496976202453060037807522, −4.44956608790212105882203027814, −3.98375307145101670305929932306, −3.97851471962543880594130770281, −3.94389515701335131694057178352, −3.64394913070110952399276718580, −3.33550101485453386116903008289, −2.97064759651776962625400815123, −2.84011768945163756429693933205, −2.48917894556172260893005336881, −2.39958354495418204134704667338, −2.24979900847689935477682389354, −1.59646771079412109620937701749, −1.49788453221545786838645193554, −1.29618632104580819877361630414, −1.12405357878212052987835012073, −0.50673974312971847292206995767, 0.50673974312971847292206995767, 1.12405357878212052987835012073, 1.29618632104580819877361630414, 1.49788453221545786838645193554, 1.59646771079412109620937701749, 2.24979900847689935477682389354, 2.39958354495418204134704667338, 2.48917894556172260893005336881, 2.84011768945163756429693933205, 2.97064759651776962625400815123, 3.33550101485453386116903008289, 3.64394913070110952399276718580, 3.94389515701335131694057178352, 3.97851471962543880594130770281, 3.98375307145101670305929932306, 4.44956608790212105882203027814, 4.66629496976202453060037807522, 4.68206754014833178583060037693, 4.84211575406691721862244066989, 5.28886286155025898388959890409, 5.29037926303853323696174108194, 5.51702390320995492468018590750, 5.81441181110851486691691011909, 6.09975052674398827965950187913, 6.15302193392129514545646118067

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