Properties

Label 8-33e8-1.1-c1e4-0-0
Degree $8$
Conductor $1.406\times 10^{12}$
Sign $1$
Analytic cond. $5717.68$
Root an. cond. $2.94884$
Motivic weight $1$
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  − 2·4-s − 5·16-s − 24·17-s + 8·25-s + 12·29-s + 24·31-s − 12·37-s − 12·41-s + 12·49-s + 20·64-s − 24·67-s + 48·68-s + 24·83-s + 4·97-s − 16·100-s − 24·101-s + 16·103-s − 48·107-s − 24·116-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 24·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s − 5/4·16-s − 5.82·17-s + 8/5·25-s + 2.22·29-s + 4.31·31-s − 1.97·37-s − 1.87·41-s + 12/7·49-s + 5/2·64-s − 2.93·67-s + 5.82·68-s + 2.63·83-s + 0.406·97-s − 8/5·100-s − 2.38·101-s + 1.57·103-s − 4.64·107-s − 2.22·116-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(5717.68\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7760025869\)
\(L(\frac12)\) \(\approx\) \(0.7760025869\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^3$ \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^3$ \( 1 + 191 T^{4} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 44 T^{2} + 1110 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
41$D_{4}$ \( ( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 60 T^{2} + 4166 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 200 T^{2} + 15591 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 188 T^{2} + 15366 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 120 T^{2} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 236 T^{2} + 23574 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 240 T^{2} + 24866 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 204 T^{2} + 19814 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 200 T^{2} + 24519 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
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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

−6.90647508025969133230113731751, −6.85984687166063966580236487388, −6.56176378712241241765738719266, −6.45159988052906533380015911926, −6.44179056610049615899471331368, −6.18032686032673888017648386882, −5.71074907123516689322783993532, −5.21180462925419925387293501956, −5.17325322263442369966679007828, −4.83543706118873890196857846015, −4.59007349984300819599575498305, −4.58264774062851477234635532610, −4.40051112998559859901954285824, −4.16961268444196018523884547604, −4.12296473446985933711398413822, −3.52368212485332414094893960270, −3.06540398965895110676113017259, −2.87432392088746124546778653490, −2.57982544613269483836840094593, −2.41132871686894256162395272253, −2.16465712856414248875942825033, −1.76808510008817437418842572621, −1.28091696158996205627575033883, −0.62732490268360665978152297161, −0.29403336948266406174791811267, 0.29403336948266406174791811267, 0.62732490268360665978152297161, 1.28091696158996205627575033883, 1.76808510008817437418842572621, 2.16465712856414248875942825033, 2.41132871686894256162395272253, 2.57982544613269483836840094593, 2.87432392088746124546778653490, 3.06540398965895110676113017259, 3.52368212485332414094893960270, 4.12296473446985933711398413822, 4.16961268444196018523884547604, 4.40051112998559859901954285824, 4.58264774062851477234635532610, 4.59007349984300819599575498305, 4.83543706118873890196857846015, 5.17325322263442369966679007828, 5.21180462925419925387293501956, 5.71074907123516689322783993532, 6.18032686032673888017648386882, 6.44179056610049615899471331368, 6.45159988052906533380015911926, 6.56176378712241241765738719266, 6.85984687166063966580236487388, 6.90647508025969133230113731751

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