Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 8·13-s − 3·16-s + 10·25-s − 16·37-s − 2·49-s − 8·52-s − 16·61-s + 7·64-s + 20·73-s + 44·97-s − 10·100-s − 16·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.21·13-s − 3/4·16-s + 2·25-s − 2.63·37-s − 2/7·49-s − 1.10·52-s − 2.04·61-s + 7/8·64-s + 2.34·73-s + 4.46·97-s − 100-s − 1.53·109-s − 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $0.964404$
$L(\frac12)$  $\approx$  $0.964404$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 101 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.45275716925881041819960738904, −9.685870156326968567023733517772, −9.519585553092712041510098831019, −9.126766705999112045448516538687, −9.102591024841214523748642571346, −8.605039139736456510941004027840, −8.538632563341344523611852273051, −8.201124739732859630903453769177, −8.034496553199522573133678458453, −7.35020195338513262000920046682, −7.06768930669604369158877238253, −7.03079453687674482395396872795, −6.27211017914488672877122195994, −6.24844149227236302898293430355, −6.16214209540224520188128453925, −5.28446112990804827637455560910, −5.16674222557969672828772180192, −4.85330436059790058789154442370, −4.46019734629326974911101879499, −3.83899928371328895260030976330, −3.59964926771827505759024991550, −3.29784359895666266011650792700, −2.66172949639314995069071382847, −1.90617931312276687744763662728, −1.18580599730778824673139974474, 1.18580599730778824673139974474, 1.90617931312276687744763662728, 2.66172949639314995069071382847, 3.29784359895666266011650792700, 3.59964926771827505759024991550, 3.83899928371328895260030976330, 4.46019734629326974911101879499, 4.85330436059790058789154442370, 5.16674222557969672828772180192, 5.28446112990804827637455560910, 6.16214209540224520188128453925, 6.24844149227236302898293430355, 6.27211017914488672877122195994, 7.03079453687674482395396872795, 7.06768930669604369158877238253, 7.35020195338513262000920046682, 8.034496553199522573133678458453, 8.201124739732859630903453769177, 8.538632563341344523611852273051, 8.605039139736456510941004027840, 9.102591024841214523748642571346, 9.126766705999112045448516538687, 9.519585553092712041510098831019, 9.685870156326968567023733517772, 10.45275716925881041819960738904

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