Properties

Label 8-105e4-1.1-c3e4-0-0
Degree $8$
Conductor $121550625$
Sign $1$
Analytic cond. $1473.06$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s − 26·9-s + 640·16-s − 250·25-s + 832·36-s + 686·49-s − 1.02e4·64-s + 944·79-s − 53·81-s + 8.00e3·100-s − 9.06e3·109-s + 5.04e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.66e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.54e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4·4-s − 0.962·9-s + 10·16-s − 2·25-s + 3.85·36-s + 2·49-s − 20·64-s + 1.34·79-s − 0.0727·81-s + 8·100-s − 7.96·109-s + 3.78·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 9.62·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.52·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1473.06\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{105} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.03079103356\)
\(L(\frac12)\) \(\approx\) \(0.03079103356\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 26 T^{2} + p^{6} T^{4} \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
11$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2}( 1 + 72 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - 2774 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 754 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{2}( 1 + 54 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 175646 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
71$C_2$ \( ( 1 - 828 T + p^{3} T^{2} )^{2}( 1 + 828 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( ( 1 - 504254 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 236 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 1141306 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + 897874 T^{2} + p^{6} T^{4} )^{2} \)
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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

−9.533737129037703884664802627089, −9.391392839877074209176750872784, −8.947711014294686640449541922188, −8.941373085706403648205544576245, −8.675200589701545521645104493429, −8.115709256405823213106949219484, −8.083948643330592681195056613167, −7.963437349696721112276010922968, −7.52709592562571170017705975291, −7.10250025143805530197075874098, −6.48505104499050042421961108682, −5.95013541487232629109365514668, −5.81046875853650777734471614348, −5.43131296013000440657097460426, −5.32900118313901463602575859670, −4.83257541359668209823974145049, −4.58951868927136206353080724687, −4.00620415773054123544866592204, −3.91598681626432828455324155804, −3.70366538254294274806929077678, −3.09800503779469979558425592120, −2.51226379568026950090055357181, −1.42538772465813365357571217180, −0.824560590213629432363638603045, −0.079474028909183583181871582819, 0.079474028909183583181871582819, 0.824560590213629432363638603045, 1.42538772465813365357571217180, 2.51226379568026950090055357181, 3.09800503779469979558425592120, 3.70366538254294274806929077678, 3.91598681626432828455324155804, 4.00620415773054123544866592204, 4.58951868927136206353080724687, 4.83257541359668209823974145049, 5.32900118313901463602575859670, 5.43131296013000440657097460426, 5.81046875853650777734471614348, 5.95013541487232629109365514668, 6.48505104499050042421961108682, 7.10250025143805530197075874098, 7.52709592562571170017705975291, 7.963437349696721112276010922968, 8.083948643330592681195056613167, 8.115709256405823213106949219484, 8.675200589701545521645104493429, 8.941373085706403648205544576245, 8.947711014294686640449541922188, 9.391392839877074209176750872784, 9.533737129037703884664802627089

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