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Properties

Label 8-3060e4-1.1-c0e4-0-12
Degree $8$
Conductor $8.768\times 10^{13}$
Sign $1$
Analytic cond. $5.43893$
Root an. cond. $1.23577$
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  − 4·2-s + 10·4-s − 20·8-s + 35·16-s − 56·32-s + 4·41-s + 4·61-s + 84·64-s − 4·73-s − 16·82-s − 4·97-s − 16·122-s + 127-s − 120·128-s + 131-s + 137-s + 139-s + 16·146-s + 149-s + 151-s + 157-s + 163-s + 40·164-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4·2-s + 10·4-s − 20·8-s + 35·16-s − 56·32-s + 4·41-s + 4·61-s + 84·64-s − 4·73-s − 16·82-s − 4·97-s − 16·122-s + 127-s − 120·128-s + 131-s + 137-s + 139-s + 16·146-s + 149-s + 151-s + 157-s + 163-s + 40·164-s + 167-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(5.43893\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2632732778\)
\(L(\frac12)\) \(\approx\) \(0.2632732778\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{4} \)
3 \( 1 \)
5$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
good7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
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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

−6.47373413867048422356631073824, −6.37289848549930400426223933687, −6.01393208395888358249790899408, −5.93481657371631230101812632346, −5.70407934122156048172397889226, −5.68653447243802228914580749557, −5.45831184823570248006052428398, −5.02769963968701127980009477051, −4.99381291142054941859876183487, −4.25608089437408820520761829403, −4.18828776709178147626351112332, −4.03295857177820524124202835959, −3.84227292391476725361307401179, −3.28510394514524721693767955953, −3.12897326743764800306754068134, −2.99537397880956825424067657657, −2.66871151928545453515410109746, −2.51774396371428459030250779395, −2.34196207798860544566049168437, −1.98897164023106547417630342083, −1.77195632754358503412352877187, −1.40071388022122075757261782379, −1.21640990348348868459427235847, −0.73483906327793830158270537435, −0.58787303426635366882585143981, 0.58787303426635366882585143981, 0.73483906327793830158270537435, 1.21640990348348868459427235847, 1.40071388022122075757261782379, 1.77195632754358503412352877187, 1.98897164023106547417630342083, 2.34196207798860544566049168437, 2.51774396371428459030250779395, 2.66871151928545453515410109746, 2.99537397880956825424067657657, 3.12897326743764800306754068134, 3.28510394514524721693767955953, 3.84227292391476725361307401179, 4.03295857177820524124202835959, 4.18828776709178147626351112332, 4.25608089437408820520761829403, 4.99381291142054941859876183487, 5.02769963968701127980009477051, 5.45831184823570248006052428398, 5.68653447243802228914580749557, 5.70407934122156048172397889226, 5.93481657371631230101812632346, 6.01393208395888358249790899408, 6.37289848549930400426223933687, 6.47373413867048422356631073824

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