Properties

Label 8-320e4-1.1-c2e4-0-3
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $5780.16$
Root an. cond. $2.95285$
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  + 16·9-s + 16·13-s − 24·17-s + 10·25-s + 8·29-s − 16·37-s − 112·41-s + 96·49-s + 176·53-s − 128·61-s + 264·73-s + 50·81-s − 88·89-s − 264·97-s − 328·101-s + 128·109-s + 504·113-s + 256·117-s + 84·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 384·153-s + 157-s + ⋯
L(s)  = 1  + 16/9·9-s + 1.23·13-s − 1.41·17-s + 2/5·25-s + 8/29·29-s − 0.432·37-s − 2.73·41-s + 1.95·49-s + 3.32·53-s − 2.09·61-s + 3.61·73-s + 0.617·81-s − 0.988·89-s − 2.72·97-s − 3.24·101-s + 1.17·109-s + 4.46·113-s + 2.18·117-s + 0.694·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 2.50·153-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5780.16\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.433864947\)
\(L(\frac12)\) \(\approx\) \(3.433864947\)
\(L(2)\) 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 \( 1 \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2^2:C_4$ \( 1 - 16 T^{2} + 206 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \)
7$C_2^2:C_4$ \( 1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2:C_4$ \( 1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 8 T + 334 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 12 T + 294 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \)
23$C_2^2:C_4$ \( 1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 1606 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 8 T + 2254 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 56 T + 3966 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$C_2^2:C_4$ \( 1 - 6896 T^{2} + 18665806 T^{4} - 6896 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2:C_4$ \( 1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 88 T + 7054 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 64 T + 5086 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 - 7536 T^{2} + 47276046 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^2:C_4$ \( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 132 T + 9894 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2:C_4$ \( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2:C_4$ \( 1 - 21296 T^{2} + 201120526 T^{4} - 21296 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 44 T + 8326 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−8.451578601633214421155859003268, −7.917032670545448728508925336146, −7.70873577238893634256152387371, −7.39337877714976062220643810942, −7.05420268709369504086731114474, −6.98799499083442646050864203757, −6.71069908052304013691699814495, −6.51963253962385560731232422003, −6.34347873748410983685767497535, −5.72102826704691937217778603235, −5.55866043077923825439788431593, −5.47213909694509482470300960254, −4.80447769971993259211372658725, −4.76683187146510693734183114139, −4.37934627267238151441255396734, −4.10537406850871062200977991313, −3.80240239140930554536661662757, −3.57646773042693015747548752794, −3.25949058607515899923515365180, −2.54356508116371214221037638935, −2.41804115093982427079582370547, −1.87010954949121341654638922819, −1.45820388001025216795132040605, −1.10978418672310627673152156832, −0.44881164330531768164790096116, 0.44881164330531768164790096116, 1.10978418672310627673152156832, 1.45820388001025216795132040605, 1.87010954949121341654638922819, 2.41804115093982427079582370547, 2.54356508116371214221037638935, 3.25949058607515899923515365180, 3.57646773042693015747548752794, 3.80240239140930554536661662757, 4.10537406850871062200977991313, 4.37934627267238151441255396734, 4.76683187146510693734183114139, 4.80447769971993259211372658725, 5.47213909694509482470300960254, 5.55866043077923825439788431593, 5.72102826704691937217778603235, 6.34347873748410983685767497535, 6.51963253962385560731232422003, 6.71069908052304013691699814495, 6.98799499083442646050864203757, 7.05420268709369504086731114474, 7.39337877714976062220643810942, 7.70873577238893634256152387371, 7.917032670545448728508925336146, 8.451578601633214421155859003268

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