Properties

Label 8-1680e4-1.1-c1e4-0-6
Degree $8$
Conductor $7.966\times 10^{12}$
Sign $1$
Analytic cond. $32385.1$
Root an. cond. $3.66263$
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  − 2·9-s + 8·11-s + 8·19-s + 10·25-s + 16·29-s − 16·31-s − 32·41-s − 2·49-s − 32·61-s − 32·71-s + 3·81-s + 16·89-s − 16·99-s − 24·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s − 16·171-s + 173-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.41·11-s + 1.83·19-s + 2·25-s + 2.97·29-s − 2.87·31-s − 4.99·41-s − 2/7·49-s − 4.09·61-s − 3.79·71-s + 1/3·81-s + 1.69·89-s − 1.60·99-s − 2.29·109-s − 0.363·121-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 + 0.923·169-s − 1.22·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(32385.1\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.653024417\)
\(L(\frac12)\) \(\approx\) \(1.653024417\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_4$ \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
41$C_4$ \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_4$ \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \)
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

−6.67427514917933090639374342942, −6.62754550826837263948662540680, −6.44648988037086414665311803493, −6.02589171913680573336794383658, −5.69562896916051872321719933051, −5.60666589712115886154968881518, −5.40008369206042728836092980195, −5.21822241519116633952655098494, −4.84800183149000539712994214537, −4.59429385487275223150575003965, −4.59066596833036635818773195792, −4.36479531582599764469203548637, −3.96408555542793988421582208168, −3.66964663581635087881691048687, −3.32299895968630026163813232651, −3.14043716612225681638802522076, −3.11286114229709259824528097070, −3.07479837211408391179632462678, −2.55117428671047054305860631940, −1.98393875322555708005438014281, −1.67740086912831011580957148508, −1.48169200962818908037988918695, −1.31107122907419097395558461800, −0.950268170574869678856090109351, −0.23296643099176709284440640549, 0.23296643099176709284440640549, 0.950268170574869678856090109351, 1.31107122907419097395558461800, 1.48169200962818908037988918695, 1.67740086912831011580957148508, 1.98393875322555708005438014281, 2.55117428671047054305860631940, 3.07479837211408391179632462678, 3.11286114229709259824528097070, 3.14043716612225681638802522076, 3.32299895968630026163813232651, 3.66964663581635087881691048687, 3.96408555542793988421582208168, 4.36479531582599764469203548637, 4.59066596833036635818773195792, 4.59429385487275223150575003965, 4.84800183149000539712994214537, 5.21822241519116633952655098494, 5.40008369206042728836092980195, 5.60666589712115886154968881518, 5.69562896916051872321719933051, 6.02589171913680573336794383658, 6.44648988037086414665311803493, 6.62754550826837263948662540680, 6.67427514917933090639374342942

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