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Properties

Label 8-330e4-1.1-c1e4-0-5
Degree $8$
Conductor $11859210000$
Sign $1$
Analytic cond. $48.2130$
Root an. cond. $1.62328$
Motivic weight $1$
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  − 2-s + 3-s + 5-s − 6-s − 4·7-s − 10-s + 11-s − 2·13-s + 4·14-s + 15-s + 13·17-s + 6·19-s − 4·21-s − 22-s + 14·23-s + 2·26-s − 29-s − 30-s + 32-s + 33-s − 13·34-s − 4·35-s + 10·37-s − 6·38-s − 2·39-s + 6·41-s + 4·42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 3.15·17-s + 1.37·19-s − 0.872·21-s − 0.213·22-s + 2.91·23-s + 0.392·26-s − 0.185·29-s − 0.182·30-s + 0.176·32-s + 0.174·33-s − 2.22·34-s − 0.676·35-s + 1.64·37-s − 0.973·38-s − 0.320·39-s + 0.937·41-s + 0.617·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(48.2130\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{330} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.923359281\)
\(L(\frac12)\) \(\approx\) \(1.923359281\)
\(L(\frac{3}{2})\) 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)$
bad2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
11$C_4$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
good7$C_2^2:C_4$ \( 1 + 4 T + 9 T^{2} + 38 T^{3} + 149 T^{4} + 38 p T^{5} + 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 16 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 - 13 T + 77 T^{2} - 355 T^{3} + 1556 T^{4} - 355 p T^{5} + 77 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 - 6 T - 3 T^{2} + 22 T^{3} + 225 T^{4} + 22 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + T - 23 T^{2} + 83 T^{3} + 900 T^{4} + 83 p T^{5} - 23 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 - 21 T^{2} + 130 T^{3} + 831 T^{4} + 130 p T^{5} - 21 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 - 10 T + 23 T^{2} + 280 T^{3} - 3191 T^{4} + 280 p T^{5} + 23 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 - 6 T - 25 T^{2} + 66 T^{3} + 1369 T^{4} + 66 p T^{5} - 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 11 T + 85 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 21 T + 149 T^{2} + 357 T^{3} + 4 T^{4} + 357 p T^{5} + 149 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 14 T + 23 T^{2} - 400 T^{3} - 2899 T^{4} - 400 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 4 T + 107 T^{2} - 142 T^{3} + 6755 T^{4} - 142 p T^{5} + 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_4\times C_2$ \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 + 16 T + 25 T^{2} - 916 T^{3} - 9471 T^{4} - 916 p T^{5} + 25 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_4\times C_2$ \( 1 - 30 T + 467 T^{2} - 5400 T^{3} + 50869 T^{4} - 5400 p T^{5} + 467 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 11 T + 17 T^{2} - 643 T^{3} + 11980 T^{4} - 643 p T^{5} + 17 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
83$C_4\times C_2$ \( 1 + 4 T - 67 T^{2} - 600 T^{3} + 3161 T^{4} - 600 p T^{5} - 67 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 + 38 T + 647 T^{2} + 6970 T^{3} + 65881 T^{4} + 6970 p T^{5} + 647 p^{2} T^{6} + 38 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.346807098200605864266798744909, −7.965956772469849319884819388251, −7.925335912575709165486973319990, −7.79286147304935546480815734430, −7.58006281429738948417772431548, −7.21812875903029804720616546149, −6.85790833163126549819217649238, −6.79453812073690855823578665174, −6.19170422630343967325151238835, −6.13282115157734824177470152705, −6.12970518136246865667113092510, −5.33554577798137018527696910163, −5.20719914704941253733190087673, −5.16966600708629897219699352910, −4.91243239663821299967650822777, −4.17706286804486207038170550852, −3.82082453585937748784114443466, −3.70997624221636420694094543977, −3.11140913679778807329830191461, −2.98300796374035956158616815583, −2.83195699459143473572801186695, −2.51661609414376649462293063011, −1.61807835429464512134003720748, −0.970268837371295582242613531242, −0.936447767184423893887300194954, 0.936447767184423893887300194954, 0.970268837371295582242613531242, 1.61807835429464512134003720748, 2.51661609414376649462293063011, 2.83195699459143473572801186695, 2.98300796374035956158616815583, 3.11140913679778807329830191461, 3.70997624221636420694094543977, 3.82082453585937748784114443466, 4.17706286804486207038170550852, 4.91243239663821299967650822777, 5.16966600708629897219699352910, 5.20719914704941253733190087673, 5.33554577798137018527696910163, 6.12970518136246865667113092510, 6.13282115157734824177470152705, 6.19170422630343967325151238835, 6.79453812073690855823578665174, 6.85790833163126549819217649238, 7.21812875903029804720616546149, 7.58006281429738948417772431548, 7.79286147304935546480815734430, 7.925335912575709165486973319990, 7.965956772469849319884819388251, 8.346807098200605864266798744909

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