Properties

Label 8-63e8-1.1-c1e4-0-6
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.00886\times 10^{6}$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 4·16-s − 20·25-s − 12·37-s − 24·43-s − 9·64-s − 8·67-s − 16·79-s + 60·100-s − 72·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 72·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3/2·4-s + 16-s − 4·25-s − 1.97·37-s − 3.65·43-s − 9/8·64-s − 0.977·67-s − 1.80·79-s + 6·100-s − 6.89·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 5.48·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.00886\times 10^{6}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^3$ \( 1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^3$ \( 1 - 114 T^{2} + 7955 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−6.28625730064740959887235522715, −6.21766688349591193440808180788, −5.98446153688585560844118214729, −5.65386427493797443469320527084, −5.52336271523020168941915072703, −5.21325837875908504152486703940, −5.18901356688970982076309928740, −5.16762747091273900619611548079, −4.87898586020754082598129614038, −4.37537709022199179604942745837, −4.36126038913861218379494643212, −4.19566804354897411512118654984, −4.09614827787021053948283933710, −3.59700804887707092832248148998, −3.55973404828077194567090508261, −3.49673628288320842682282396417, −3.33056418207869259903208340612, −2.73454580224607483850903244392, −2.65201577598597211162084295258, −2.40844361335285914984531668345, −1.96418117660714103075836856817, −1.90029272559669778304594296788, −1.35983003579650285762811723788, −1.33745529344424522133029828183, −1.23030959639264216350262426846, 0, 0, 0, 0, 1.23030959639264216350262426846, 1.33745529344424522133029828183, 1.35983003579650285762811723788, 1.90029272559669778304594296788, 1.96418117660714103075836856817, 2.40844361335285914984531668345, 2.65201577598597211162084295258, 2.73454580224607483850903244392, 3.33056418207869259903208340612, 3.49673628288320842682282396417, 3.55973404828077194567090508261, 3.59700804887707092832248148998, 4.09614827787021053948283933710, 4.19566804354897411512118654984, 4.36126038913861218379494643212, 4.37537709022199179604942745837, 4.87898586020754082598129614038, 5.16762747091273900619611548079, 5.18901356688970982076309928740, 5.21325837875908504152486703940, 5.52336271523020168941915072703, 5.65386427493797443469320527084, 5.98446153688585560844118214729, 6.21766688349591193440808180788, 6.28625730064740959887235522715

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