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Properties

Label 8-3120e4-1.1-c0e4-0-0
Degree $8$
Conductor $9.476\times 10^{13}$
Sign $1$
Analytic cond. $5.87823$
Root an. cond. $1.24783$
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 − 4·13-s − 16-s − 4·61-s − 8·79-s + 3·81-s + 4·103-s + 8·117-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + ⋯
L(s)  = 1  − 2·9-s − 4·13-s − 16-s − 4·61-s − 8·79-s + 3·81-s + 4·103-s + 8·117-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(5.87823\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005383930850\)
\(L(\frac12)\) \(\approx\) \(0.005383930850\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$ \( ( 1 + T )^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.21687650180838788583029712927, −6.09298793381735070424610294119, −6.02247436752145706597905117965, −5.72245124528145281391761941462, −5.46618164495475627925674743316, −5.37228229604546389526875704777, −5.02497559814467665523069253253, −4.93948764310188089129296910328, −4.89881529778348761917045359375, −4.47689487045865461026577282405, −4.31126464811834611174770933723, −4.21357507361282376990810345026, −4.09074671647151457808933431874, −3.39969180934933823074972378983, −3.29711769106704733293278413975, −3.15133503387050127331573933314, −2.66877223551908649558626449409, −2.62083761514459993846421886758, −2.59755952013263953890126295506, −2.53318429246720296807905275780, −1.79181793040799001112143620030, −1.67382059304371206078745496482, −1.60738948439222232303369630040, −0.73174698881786034115263731854, −0.03470692101782975450575558167, 0.03470692101782975450575558167, 0.73174698881786034115263731854, 1.60738948439222232303369630040, 1.67382059304371206078745496482, 1.79181793040799001112143620030, 2.53318429246720296807905275780, 2.59755952013263953890126295506, 2.62083761514459993846421886758, 2.66877223551908649558626449409, 3.15133503387050127331573933314, 3.29711769106704733293278413975, 3.39969180934933823074972378983, 4.09074671647151457808933431874, 4.21357507361282376990810345026, 4.31126464811834611174770933723, 4.47689487045865461026577282405, 4.89881529778348761917045359375, 4.93948764310188089129296910328, 5.02497559814467665523069253253, 5.37228229604546389526875704777, 5.46618164495475627925674743316, 5.72245124528145281391761941462, 6.02247436752145706597905117965, 6.09298793381735070424610294119, 6.21687650180838788583029712927

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