Properties

Label 8-28e8-1.1-c1e4-0-3
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s − 4·11-s − 8·23-s + 2·25-s + 8·29-s − 20·37-s − 8·43-s + 4·53-s + 24·67-s + 48·71-s − 8·79-s + 9·81-s − 16·99-s − 8·107-s + 4·109-s − 48·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + ⋯
L(s)  = 1  + 4/3·9-s − 1.20·11-s − 1.66·23-s + 2/5·25-s + 1.48·29-s − 3.28·37-s − 1.21·43-s + 0.549·53-s + 2.93·67-s + 5.69·71-s − 0.900·79-s + 81-s − 1.60·99-s − 0.773·107-s + 0.383·109-s − 4.51·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-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: induced by $\chi_{784} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(1.574331435\)
\(L(\frac12)\) \(\approx\) \(1.574331435\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^3$ \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^3$ \( 1 + 12 T^{2} - 217 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 + 10 T^{2} - 861 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 114 T^{2} + 9275 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.62074525402199612727827071418, −7.00871055966131451992737695214, −6.88151375599204877770571602050, −6.67075843762315298985100363056, −6.52300640356107246176646557665, −6.45247403005873939319753723120, −6.18304243457929548922066132055, −5.51380781264771516297328365553, −5.41331540450567820387993550142, −5.17782408666456329625649290108, −5.14997551086115392925216488356, −4.85354118211895599616948036798, −4.70530017664041149978820721322, −3.94012971856354852972276658464, −3.89662370995756469430218815986, −3.82060634655989023985418983997, −3.70207059256534126130445520889, −3.12069013109595285993598872919, −2.63833122172139736969369907396, −2.53548501848702022901603781564, −2.24897778089507605317827534856, −1.73243044668630488788654761720, −1.54867837683605803585089452534, −1.00846479455911153282846250281, −0.34433996568311078907313493030, 0.34433996568311078907313493030, 1.00846479455911153282846250281, 1.54867837683605803585089452534, 1.73243044668630488788654761720, 2.24897778089507605317827534856, 2.53548501848702022901603781564, 2.63833122172139736969369907396, 3.12069013109595285993598872919, 3.70207059256534126130445520889, 3.82060634655989023985418983997, 3.89662370995756469430218815986, 3.94012971856354852972276658464, 4.70530017664041149978820721322, 4.85354118211895599616948036798, 5.14997551086115392925216488356, 5.17782408666456329625649290108, 5.41331540450567820387993550142, 5.51380781264771516297328365553, 6.18304243457929548922066132055, 6.45247403005873939319753723120, 6.52300640356107246176646557665, 6.67075843762315298985100363056, 6.88151375599204877770571602050, 7.00871055966131451992737695214, 7.62074525402199612727827071418

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