Properties

Label 8-2790e4-1.1-c1e4-0-0
Degree $8$
Conductor $6.059\times 10^{13}$
Sign $1$
Analytic cond. $246334.$
Root an. cond. $4.71998$
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 − 8·5-s − 8·11-s + 3·16-s + 16·20-s + 38·25-s + 16·29-s + 4·31-s − 8·41-s + 16·44-s + 4·49-s + 64·55-s + 24·59-s − 8·61-s − 4·64-s − 24·80-s + 8·89-s − 76·100-s + 24·109-s − 32·116-s + 12·121-s − 8·124-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s − 3.57·5-s − 2.41·11-s + 3/4·16-s + 3.57·20-s + 38/5·25-s + 2.97·29-s + 0.718·31-s − 1.24·41-s + 2.41·44-s + 4/7·49-s + 8.62·55-s + 3.12·59-s − 1.02·61-s − 1/2·64-s − 2.68·80-s + 0.847·89-s − 7.59·100-s + 2.29·109-s − 2.97·116-s + 1.09·121-s − 0.718·124-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(246334.\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6187256129\)
\(L(\frac12)\) \(\approx\) \(0.6187256129\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_1$ \( ( 1 - T )^{4} \)
good7$C_4\times C_2$ \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 4 T^{2} + 1590 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 36 T^{2} + 4694 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 100 T^{2} + 4966 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 236 T^{2} + 25654 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.28490646401135145631564581838, −5.98340164432540046319805181445, −5.91200830580157344034137548353, −5.32336525289576685220608852465, −5.25325373515817224139226357954, −5.05201035346064229935749506238, −5.05171863136366667819298307792, −4.81532432350865930193125161278, −4.46290480656056156750578325378, −4.38254680231810686805080187457, −4.27180711269304897947265671847, −3.85565269632063985398092528753, −3.77576094433654918868844930838, −3.48353639321229734455327677952, −3.46503508504033697673268738029, −2.83884163364883867871465010789, −2.81792248649561282165864562735, −2.73213947864518013787378141391, −2.63434620837729844268528326940, −1.93184028240536928334888804179, −1.66357846160614310160657713703, −0.952230693505777515026116116314, −0.846955715637141101927702635843, −0.51147160771412702295183990121, −0.28521836138897651654208309530, 0.28521836138897651654208309530, 0.51147160771412702295183990121, 0.846955715637141101927702635843, 0.952230693505777515026116116314, 1.66357846160614310160657713703, 1.93184028240536928334888804179, 2.63434620837729844268528326940, 2.73213947864518013787378141391, 2.81792248649561282165864562735, 2.83884163364883867871465010789, 3.46503508504033697673268738029, 3.48353639321229734455327677952, 3.77576094433654918868844930838, 3.85565269632063985398092528753, 4.27180711269304897947265671847, 4.38254680231810686805080187457, 4.46290480656056156750578325378, 4.81532432350865930193125161278, 5.05171863136366667819298307792, 5.05201035346064229935749506238, 5.25325373515817224139226357954, 5.32336525289576685220608852465, 5.91200830580157344034137548353, 5.98340164432540046319805181445, 6.28490646401135145631564581838

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