Properties

Label 8-39e8-1.1-c1e4-0-3
Degree $8$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $21758.3$
Root an. cond. $3.48500$
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 − 10·25-s − 8·43-s − 4·49-s − 28·61-s − 20·64-s + 32·79-s − 20·100-s − 8·103-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 8·196-s + ⋯
L(s)  = 1  + 4-s − 5/4·16-s − 2·25-s − 1.21·43-s − 4/7·49-s − 3.58·61-s − 5/2·64-s + 3.60·79-s − 2·100-s − 0.788·103-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 − 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4/7·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(21758.3\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.987488796\)
\(L(\frac12)\) \(\approx\) \(1.987488796\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 71 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 77 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.63339800447605068874570904626, −6.57650722137274033273145283721, −6.54000410023318868922375130505, −6.11935604460953131572433872004, −6.08368432190686119347373147348, −5.58706080568863421411953848254, −5.39951484836429333868404903685, −5.26871767392059862585718345662, −5.23316109809968390773540466347, −4.56188853449087200507201522328, −4.43922427981244934972207506445, −4.35924213581238394881552865739, −4.22225059875039158251176681207, −3.87035595160793097875695030727, −3.33935580569889001720223598412, −3.18262028694788129275737234824, −3.12247599941473213509210880389, −2.92602096648986969988191730383, −2.28831522888973962679995276751, −2.12319073653516606276439508814, −1.93693266734966616326600412945, −1.75473925637323638732848827979, −1.42111218525009464563289843159, −0.70482319783376986506171929457, −0.29872830869381274131204375417, 0.29872830869381274131204375417, 0.70482319783376986506171929457, 1.42111218525009464563289843159, 1.75473925637323638732848827979, 1.93693266734966616326600412945, 2.12319073653516606276439508814, 2.28831522888973962679995276751, 2.92602096648986969988191730383, 3.12247599941473213509210880389, 3.18262028694788129275737234824, 3.33935580569889001720223598412, 3.87035595160793097875695030727, 4.22225059875039158251176681207, 4.35924213581238394881552865739, 4.43922427981244934972207506445, 4.56188853449087200507201522328, 5.23316109809968390773540466347, 5.26871767392059862585718345662, 5.39951484836429333868404903685, 5.58706080568863421411953848254, 6.08368432190686119347373147348, 6.11935604460953131572433872004, 6.54000410023318868922375130505, 6.57650722137274033273145283721, 6.63339800447605068874570904626

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