Properties

Label 8-28e8-1.1-c1e4-0-18
Degree $8$
Conductor $377801998336$
Sign $1$
Analytic cond. $1535.93$
Root an. cond. $2.50205$
Motivic weight $1$
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  + 2·2-s + 4·3-s + 2·4-s + 4·5-s + 8·6-s + 4·8-s + 8·9-s + 8·10-s + 6·11-s + 8·12-s + 16·15-s + 8·16-s + 12·17-s + 16·18-s + 8·19-s + 8·20-s + 12·22-s + 16·24-s + 8·25-s + 8·27-s − 4·29-s + 32·30-s − 8·31-s + 8·32-s + 24·33-s + 24·34-s + 16·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 1.78·5-s + 3.26·6-s + 1.41·8-s + 8/3·9-s + 2.52·10-s + 1.80·11-s + 2.30·12-s + 4.13·15-s + 2·16-s + 2.91·17-s + 3.77·18-s + 1.83·19-s + 1.78·20-s + 2.55·22-s + 3.26·24-s + 8/5·25-s + 1.53·27-s − 0.742·29-s + 5.84·30-s − 1.43·31-s + 1.41·32-s + 4.17·33-s + 4.11·34-s + 8/3·36-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1535.93\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(41.66485489\)
\(L(\frac12)\) \(\approx\) \(41.66485489\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^3$ \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^3$ \( 1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 6 T + 18 T^{2} + 24 T^{3} - 193 T^{4} + 24 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 - 8 T + 32 T^{2} + 48 T^{3} - 553 T^{4} + 48 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2^3$ \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 336 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 14 T + 98 T^{2} - 112 T^{3} - 3593 T^{4} - 112 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^3$ \( 1 + 4 T + 8 T^{2} - 440 T^{3} - 4361 T^{4} - 440 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^3$ \( 1 + 12 T + 72 T^{2} - 600 T^{3} - 7321 T^{4} - 600 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 10 T + 50 T^{2} + 840 T^{3} - 8689 T^{4} + 840 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 142 T^{2} + 12243 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−7.33747426266891478602929240584, −7.08150784865022386570986335285, −7.07587892314230807299011682317, −6.76825585613114933817199966427, −6.38651164328875580637393767431, −6.11914992922886197742259635928, −6.08038672981610837688879027970, −5.45030257338155297736313814494, −5.43323867489039073027346539828, −5.36575435864884167771114344280, −5.15529902441222341512159799302, −4.81418769115728145002788669701, −4.45298092089365141481227496453, −4.04429507960499404210854820902, −3.74141223598490187830796492820, −3.53775452273125727170946919439, −3.49985201963631290002301364581, −3.40678795799402772404462084427, −2.79237547480323226477903455942, −2.73761536634937904127457221813, −2.29464575743850800077466704940, −1.82223302649764776808940690760, −1.44513081243145069427958275784, −1.36420564415615086607747273350, −1.31379694100316589003522712701, 1.31379694100316589003522712701, 1.36420564415615086607747273350, 1.44513081243145069427958275784, 1.82223302649764776808940690760, 2.29464575743850800077466704940, 2.73761536634937904127457221813, 2.79237547480323226477903455942, 3.40678795799402772404462084427, 3.49985201963631290002301364581, 3.53775452273125727170946919439, 3.74141223598490187830796492820, 4.04429507960499404210854820902, 4.45298092089365141481227496453, 4.81418769115728145002788669701, 5.15529902441222341512159799302, 5.36575435864884167771114344280, 5.43323867489039073027346539828, 5.45030257338155297736313814494, 6.08038672981610837688879027970, 6.11914992922886197742259635928, 6.38651164328875580637393767431, 6.76825585613114933817199966427, 7.07587892314230807299011682317, 7.08150784865022386570986335285, 7.33747426266891478602929240584

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