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Properties

Label 8-1760e4-1.1-c0e4-0-1
Degree $8$
Conductor $9.595\times 10^{12}$
Sign $1$
Analytic cond. $0.595222$
Root an. cond. $0.937205$
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  − 5-s − 3·7-s − 9-s + 11-s + 3·13-s + 2·19-s + 2·23-s + 3·35-s − 2·37-s + 3·41-s + 45-s + 2·47-s + 6·49-s − 2·53-s − 55-s + 2·59-s + 3·63-s − 3·65-s − 3·77-s − 2·89-s − 9·91-s − 2·95-s − 99-s + 2·103-s − 2·115-s − 3·117-s + 127-s + ⋯
L(s)  = 1  − 5-s − 3·7-s − 9-s + 11-s + 3·13-s + 2·19-s + 2·23-s + 3·35-s − 2·37-s + 3·41-s + 45-s + 2·47-s + 6·49-s − 2·53-s − 55-s + 2·59-s + 3·63-s − 3·65-s − 3·77-s − 2·89-s − 9·91-s − 2·95-s − 99-s + 2·103-s − 2·115-s − 3·117-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.595222\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8113675986\)
\(L(\frac12)\) \(\approx\) \(0.8113675986\)
\(L(1)\) 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 \( 1 \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
13$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
23$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−6.91422391118346260688275103098, −6.49581508316893380590510897941, −6.28414850965238338225282722669, −6.25547817823566540568812772458, −6.23843428413840603969229575990, −5.64213232100910980207937009470, −5.59196666569741998606513366905, −5.51480734744154331327818140718, −5.36880373684449116142741007375, −4.80058204970272185095811744385, −4.70777305820734083552351770854, −4.19639713228725902351165731021, −3.96936474838488143653031106608, −3.73610422671316602400263277353, −3.60220335600062792448247746966, −3.59643784152287024785961363455, −3.49957008128929798177648782320, −2.88223035855807665919701929482, −2.77919337516118202356259089717, −2.57278663030432951781979691439, −2.54201189203636174113634881132, −1.48941698097385894892291481055, −1.33371165994113460471617461485, −1.04560009404662807490430644014, −0.63229069218013483504699716298, 0.63229069218013483504699716298, 1.04560009404662807490430644014, 1.33371165994113460471617461485, 1.48941698097385894892291481055, 2.54201189203636174113634881132, 2.57278663030432951781979691439, 2.77919337516118202356259089717, 2.88223035855807665919701929482, 3.49957008128929798177648782320, 3.59643784152287024785961363455, 3.60220335600062792448247746966, 3.73610422671316602400263277353, 3.96936474838488143653031106608, 4.19639713228725902351165731021, 4.70777305820734083552351770854, 4.80058204970272185095811744385, 5.36880373684449116142741007375, 5.51480734744154331327818140718, 5.59196666569741998606513366905, 5.64213232100910980207937009470, 6.23843428413840603969229575990, 6.25547817823566540568812772458, 6.28414850965238338225282722669, 6.49581508316893380590510897941, 6.91422391118346260688275103098

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