Properties

Degree $8$
Conductor $1.032\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s − 2·7-s + 6·9-s + 4·11-s + 8·15-s + 8·17-s − 20·19-s + 8·21-s − 2·23-s + 5·25-s + 4·27-s + 4·29-s + 12·31-s − 16·33-s + 4·35-s + 8·37-s + 4·43-s − 12·45-s + 49-s − 32·51-s − 24·53-s − 8·55-s + 80·57-s − 4·59-s − 18·61-s − 12·63-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s − 0.755·7-s + 2·9-s + 1.20·11-s + 2.06·15-s + 1.94·17-s − 4.58·19-s + 1.74·21-s − 0.417·23-s + 25-s + 0.769·27-s + 0.742·29-s + 2.15·31-s − 2.78·33-s + 0.676·35-s + 1.31·37-s + 0.609·43-s − 1.78·45-s + 1/7·49-s − 4.48·51-s − 3.29·53-s − 1.07·55-s + 10.5·57-s − 0.520·59-s − 2.30·61-s − 1.51·63-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2956768265\)
\(L(\frac12)\) \(\approx\) \(0.2956768265\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 + 2 T - T^{2} - 2 p T^{3} - 4 p T^{4} - 2 p^{2} T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$D_{4}$ \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 4 T - 50 T^{2} + 80 T^{3} + 1819 T^{4} + 80 p T^{5} - 50 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^3$ \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 16 T + 82 T^{2} + 640 T^{3} + 8635 T^{4} + 640 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 6 T - 107 T^{2} + 90 T^{3} + 11364 T^{4} + 90 p T^{5} - 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.08220885182221980249453396425, −6.60946371017500205569420558912, −6.59346717162850579785892586816, −6.44139223218738944001638154222, −6.23042405133499405351376482239, −5.95048934908979591808701958876, −5.88901167290752387832275338558, −5.76642568124034498705433522459, −5.62641154372072107758146933498, −4.78373234621885571599501331940, −4.67130306694046274869808185284, −4.58615775155479881587904839052, −4.44962215031996023637117947626, −4.31063365361766242026333753990, −4.19795613890689535319049717561, −3.35592605166668411807361210881, −3.25859243012756859283470059073, −3.08851703960304538358922525791, −2.96158404149998557851368046115, −2.25013061593942333521244211182, −2.06120870414923692069032507984, −1.32586673763996818821308854854, −1.31698624389686944890217139504, −0.60164079668848799712035087560, −0.25481528615209441411258965761, 0.25481528615209441411258965761, 0.60164079668848799712035087560, 1.31698624389686944890217139504, 1.32586673763996818821308854854, 2.06120870414923692069032507984, 2.25013061593942333521244211182, 2.96158404149998557851368046115, 3.08851703960304538358922525791, 3.25859243012756859283470059073, 3.35592605166668411807361210881, 4.19795613890689535319049717561, 4.31063365361766242026333753990, 4.44962215031996023637117947626, 4.58615775155479881587904839052, 4.67130306694046274869808185284, 4.78373234621885571599501331940, 5.62641154372072107758146933498, 5.76642568124034498705433522459, 5.88901167290752387832275338558, 5.95048934908979591808701958876, 6.23042405133499405351376482239, 6.44139223218738944001638154222, 6.59346717162850579785892586816, 6.60946371017500205569420558912, 7.08220885182221980249453396425

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