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Properties

Label 8-72e4-1.1-c2e4-0-3
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $14.8139$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 2·3-s + 4·4-s + 8·6-s − 16·8-s + 9·9-s + 14·11-s + 8·12-s − 64·16-s − 4·17-s + 36·18-s + 68·19-s + 56·22-s − 32·24-s − 50·25-s + 46·27-s − 64·32-s + 28·33-s − 16·34-s + 36·36-s + 272·38-s − 46·41-s + 14·43-s + 56·44-s − 128·48-s − 98·49-s − 200·50-s + ⋯
L(s)  = 1  + 2·2-s + 2/3·3-s + 4-s + 4/3·6-s − 2·8-s + 9-s + 1.27·11-s + 2/3·12-s − 4·16-s − 0.235·17-s + 2·18-s + 3.57·19-s + 2.54·22-s − 4/3·24-s − 2·25-s + 1.70·27-s − 2·32-s + 0.848·33-s − 0.470·34-s + 36-s + 7.15·38-s − 1.12·41-s + 0.325·43-s + 1.27·44-s − 8/3·48-s − 2·49-s − 4·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(14.8139\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.228816372\)
\(L(\frac12)\) \(\approx\) \(5.228816372\)
\(L(2)\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T - 5 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
good5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \)
13$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 2 T - 285 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T + 795 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
41$C_2$$\times$$C_2^2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2}( 1 - 46 T + 435 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} ) \)
43$C_2$$\times$$C_2^2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 + 14 T - 1653 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \)
47$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2}( 1 - 82 T + 3243 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} ) \)
61$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 62 T + p^{2} T^{2} )^{2}( 1 + 62 T - 645 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2^2$ \( ( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 158 T + 18075 T^{2} + 158 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 146 T + p^{2} T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 94 T + p^{2} T^{2} )^{2}( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} ) \)
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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

−10.83701988600285007599802811195, −10.13211271349233195653081717861, −9.719219838227309434069861295979, −9.707025863920982988366364925845, −9.630781934241601766184631095070, −9.207086370509510394378466322293, −8.822755435693462262838826870640, −8.549531182686802293601379642008, −8.183835410055866268691537998988, −7.57850517660214343536549862651, −7.51975771124509058498275340277, −7.17092255581944030826355125046, −6.34820666215631933030829094592, −6.31365036941979104863846833998, −6.28444123948809859784341025928, −5.38294737984831968263409884575, −5.11113809757699299506404675381, −4.98888721332793020568271603921, −4.55345684454622345965266477311, −3.82871892362638352406689144689, −3.71988799327917254607165538179, −3.39335713498662159480546664509, −2.98866063458909965765364297058, −2.16536297487933283441156866612, −1.14658148866337258030468862217, 1.14658148866337258030468862217, 2.16536297487933283441156866612, 2.98866063458909965765364297058, 3.39335713498662159480546664509, 3.71988799327917254607165538179, 3.82871892362638352406689144689, 4.55345684454622345965266477311, 4.98888721332793020568271603921, 5.11113809757699299506404675381, 5.38294737984831968263409884575, 6.28444123948809859784341025928, 6.31365036941979104863846833998, 6.34820666215631933030829094592, 7.17092255581944030826355125046, 7.51975771124509058498275340277, 7.57850517660214343536549862651, 8.183835410055866268691537998988, 8.549531182686802293601379642008, 8.822755435693462262838826870640, 9.207086370509510394378466322293, 9.630781934241601766184631095070, 9.707025863920982988366364925845, 9.719219838227309434069861295979, 10.13211271349233195653081717861, 10.83701988600285007599802811195

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