Properties

Label 8-3072e4-1.1-c0e4-0-3
Degree $8$
Conductor $8.906\times 10^{13}$
Sign $1$
Analytic cond. $5.52475$
Root an. cond. $1.23819$
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  + 4·49-s − 81-s + 8·97-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  + 4·49-s − 81-s + 8·97-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5.52475\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779017518\)
\(L(\frac12)\) \(\approx\) \(1.779017518\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
good5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_1$ \( ( 1 - T )^{8} \)
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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

−6.35987649485421005494787342589, −5.99574931940502672455653576533, −5.93490095436040219595301790049, −5.86135806230783521844654220983, −5.79916754763116127285146233624, −5.12663337476553957180311472218, −5.10259930224557558556830228620, −5.02716164288572057464889281164, −4.82184414023189294510672141893, −4.57924482306293120338808032144, −4.24438376670352310000488130924, −4.00689566264957394810232429056, −3.90294273121176848437951031786, −3.63967306527139709749155052014, −3.54815237099680412111304605985, −3.15747861379236883038058783961, −2.91612516622658256786460100480, −2.65376650948268390370063819365, −2.50229553473516403073351853980, −2.14011945446793410127392067925, −1.86676712152264104142063758650, −1.83951326644066734521503688133, −1.21653681652877569593334382239, −0.848209316173399195777955746979, −0.68821966455674584709200266864, 0.68821966455674584709200266864, 0.848209316173399195777955746979, 1.21653681652877569593334382239, 1.83951326644066734521503688133, 1.86676712152264104142063758650, 2.14011945446793410127392067925, 2.50229553473516403073351853980, 2.65376650948268390370063819365, 2.91612516622658256786460100480, 3.15747861379236883038058783961, 3.54815237099680412111304605985, 3.63967306527139709749155052014, 3.90294273121176848437951031786, 4.00689566264957394810232429056, 4.24438376670352310000488130924, 4.57924482306293120338808032144, 4.82184414023189294510672141893, 5.02716164288572057464889281164, 5.10259930224557558556830228620, 5.12663337476553957180311472218, 5.79916754763116127285146233624, 5.86135806230783521844654220983, 5.93490095436040219595301790049, 5.99574931940502672455653576533, 6.35987649485421005494787342589

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