Properties

Label 8-1200e4-1.1-c2e4-0-29
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
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  + 2·9-s − 8·19-s + 72·31-s + 124·49-s + 328·61-s + 552·79-s − 77·81-s − 152·109-s + 444·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 164·169-s − 16·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/9·9-s − 0.421·19-s + 2.32·31-s + 2.53·49-s + 5.37·61-s + 6.98·79-s − 0.950·81-s − 1.39·109-s + 3.66·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.970·169-s − 0.0935·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(8.191124729\)
\(L(\frac12)\) \(\approx\) \(8.191124729\)
\(L(2)\) 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 \( 1 \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 222 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 878 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 702 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3442 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 1998 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6942 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8402 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 5182 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 138 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 4958 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( ( 1 + 8738 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.81214907909307161164452427003, −6.57949228244667172174811423115, −6.27689069787469570655043788782, −6.13949638893133785769813951083, −6.04804253610124767334540541956, −5.54958689365246866356610242361, −5.29537194593361926368684067240, −5.25982800382105491867861793039, −5.08917522522300372601671707190, −4.72487017414647091831688352258, −4.37775575061989648671811380127, −4.30055280063726088439808907998, −3.95516148501361182646853280115, −3.68701395721638939525838239366, −3.66543254198550618174653912190, −3.21992907080917550635149665124, −2.88280021473956159449404422373, −2.62847361549277701450153541401, −2.18846041561458016478118013203, −2.14576131624126011592852865662, −2.01999778584513188836074021663, −1.26863766900372015204362167603, −0.857819559536096679552470903875, −0.71287348438050975248604658101, −0.52006480552684253072183123665, 0.52006480552684253072183123665, 0.71287348438050975248604658101, 0.857819559536096679552470903875, 1.26863766900372015204362167603, 2.01999778584513188836074021663, 2.14576131624126011592852865662, 2.18846041561458016478118013203, 2.62847361549277701450153541401, 2.88280021473956159449404422373, 3.21992907080917550635149665124, 3.66543254198550618174653912190, 3.68701395721638939525838239366, 3.95516148501361182646853280115, 4.30055280063726088439808907998, 4.37775575061989648671811380127, 4.72487017414647091831688352258, 5.08917522522300372601671707190, 5.25982800382105491867861793039, 5.29537194593361926368684067240, 5.54958689365246866356610242361, 6.04804253610124767334540541956, 6.13949638893133785769813951083, 6.27689069787469570655043788782, 6.57949228244667172174811423115, 6.81214907909307161164452427003

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