Properties

Label 8-3840e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.174\times 10^{14}$
Sign $1$
Analytic cond. $13.4881$
Root an. cond. $1.38434$
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  − 4·5-s + 10·25-s − 81-s − 4·121-s − 20·125-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 + ⋯
L(s)  = 1  − 4·5-s + 10·25-s − 81-s − 4·121-s − 20·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(13.4881\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0002968355851\)
\(L(\frac12)\) \(\approx\) \(0.0002968355851\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{4} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.22461103655919537040496537999, −6.06118736793739334712054466473, −5.78808989137020646873215763814, −5.42832202003097371252154260544, −5.40954698290835439253901618813, −5.01200135443536793589265335769, −4.88684051379152758422520559656, −4.67052556955220412421916382772, −4.61360348343072813955555176988, −4.45057210495473343033195865516, −3.96092236669099698664225630391, −3.90952483144489179310417362983, −3.81356137558899594991905410199, −3.63240466849811611118150125263, −3.48764252977294784069339095837, −3.15671669102598803735436745692, −2.96095573817742705856300078850, −2.57070507008022208528793041421, −2.47093531157134890589620554362, −2.46337935622586281392649732504, −1.68649776864335353221787410115, −1.31464935370800166507967300460, −1.12418347067374407920736451455, −0.907516935546230755221536131392, −0.008150018827572329029354412215, 0.008150018827572329029354412215, 0.907516935546230755221536131392, 1.12418347067374407920736451455, 1.31464935370800166507967300460, 1.68649776864335353221787410115, 2.46337935622586281392649732504, 2.47093531157134890589620554362, 2.57070507008022208528793041421, 2.96095573817742705856300078850, 3.15671669102598803735436745692, 3.48764252977294784069339095837, 3.63240466849811611118150125263, 3.81356137558899594991905410199, 3.90952483144489179310417362983, 3.96092236669099698664225630391, 4.45057210495473343033195865516, 4.61360348343072813955555176988, 4.67052556955220412421916382772, 4.88684051379152758422520559656, 5.01200135443536793589265335769, 5.40954698290835439253901618813, 5.42832202003097371252154260544, 5.78808989137020646873215763814, 6.06118736793739334712054466473, 6.22461103655919537040496537999

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