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Properties

Label 8-34e8-1.1-c0e4-0-0
Degree $8$
Conductor $1.786\times 10^{12}$
Sign $1$
Analytic cond. $0.110779$
Root an. cond. $0.759551$
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  − 16-s + 8·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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  − 16-s + 8·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(0.110779\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8436205050\)
\(L(\frac12)\) \(\approx\) \(0.8436205050\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
17 \( 1 \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_4\times C_2$ \( 1 + T^{8} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_4\times C_2$ \( 1 + T^{8} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_4\times C_2$ \( 1 + T^{8} \)
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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

−7.32473753966560610280228610445, −7.05791250135610464860817171784, −6.68481664619663487812150277937, −6.56293357944531347313049841148, −6.26544091862923101944558774158, −6.18874151828348691825630600809, −5.96628239236071102952080350543, −5.77285885984131962506888804467, −5.31161014283834064940063943495, −5.16635277011696563143523971531, −4.89797416652445988361001569566, −4.68533095736488509834066282519, −4.66354630754223888740597022685, −4.27584797284342223247705736725, −3.84080170621645936810794188847, −3.69877674528110823115230203257, −3.47817760586934465473963088615, −3.37186842420301803839305182207, −2.63818819517444431457608271640, −2.59063033828554983524913583898, −2.47777126484619914499070930457, −1.94000668090253920983587136554, −1.68419321045449156319243582591, −1.27108921017610214870066629693, −0.68720107346924350493028983890, 0.68720107346924350493028983890, 1.27108921017610214870066629693, 1.68419321045449156319243582591, 1.94000668090253920983587136554, 2.47777126484619914499070930457, 2.59063033828554983524913583898, 2.63818819517444431457608271640, 3.37186842420301803839305182207, 3.47817760586934465473963088615, 3.69877674528110823115230203257, 3.84080170621645936810794188847, 4.27584797284342223247705736725, 4.66354630754223888740597022685, 4.68533095736488509834066282519, 4.89797416652445988361001569566, 5.16635277011696563143523971531, 5.31161014283834064940063943495, 5.77285885984131962506888804467, 5.96628239236071102952080350543, 6.18874151828348691825630600809, 6.26544091862923101944558774158, 6.56293357944531347313049841148, 6.68481664619663487812150277937, 7.05791250135610464860817171784, 7.32473753966560610280228610445

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