Properties

Label 8-1280e4-1.1-c1e4-0-6
Degree $8$
Conductor $2.684\times 10^{12}$
Sign $1$
Analytic cond. $10913.1$
Root an. cond. $3.19700$
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  + 4·7-s + 28·23-s − 2·25-s + 8·31-s − 32·41-s − 20·47-s − 8·49-s − 8·71-s + 16·79-s + 2·81-s + 24·89-s − 48·97-s − 4·103-s + 24·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 112·161-s + 163-s + 167-s + 12·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s + 5.83·23-s − 2/5·25-s + 1.43·31-s − 4.99·41-s − 2.91·47-s − 8/7·49-s − 0.949·71-s + 1.80·79-s + 2/9·81-s + 2.54·89-s − 4.87·97-s − 0.394·103-s + 2.25·113-s + 3.27·121-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 + 8.82·161-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10913.1\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.836190293\)
\(L(\frac12)\) \(\approx\) \(2.836190293\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$D_{4}$ \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 - 14 T + 90 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
31$C_4$ \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
41$C_4$ \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 80 T^{2} + 5118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 - 164 T^{2} + 12406 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 176 T^{2} + 16542 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \)
71$C_4$ \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 224 T^{2} + 24702 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
89$C_4$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 24 T + 318 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.93349951101395621475920838167, −6.84595645193134669203456303169, −6.52114040423523938238352679120, −6.39549452819894005964596905258, −6.02171262674421941386979504299, −5.87160259812856106072880611378, −5.24971666641931078667979733510, −5.16223359089916934568800386909, −5.12932362238935979701801348881, −4.90465943014338747054474285055, −4.77573675172463568704358674363, −4.57637837066157715345068957727, −4.51232156531942864561008690813, −3.70998350958847169357695749324, −3.43044816001864398054195599184, −3.42555364820719516974219742284, −3.31860641204215112155323890171, −2.81298837348861697201773518102, −2.68971168765809430516447262763, −2.29272125026118767307977995788, −1.81908960562838514009757382627, −1.43665351424078186247477495846, −1.25242770333003207017693364841, −1.21747628466182784883573378349, −0.32937481565819254805213745795, 0.32937481565819254805213745795, 1.21747628466182784883573378349, 1.25242770333003207017693364841, 1.43665351424078186247477495846, 1.81908960562838514009757382627, 2.29272125026118767307977995788, 2.68971168765809430516447262763, 2.81298837348861697201773518102, 3.31860641204215112155323890171, 3.42555364820719516974219742284, 3.43044816001864398054195599184, 3.70998350958847169357695749324, 4.51232156531942864561008690813, 4.57637837066157715345068957727, 4.77573675172463568704358674363, 4.90465943014338747054474285055, 5.12932362238935979701801348881, 5.16223359089916934568800386909, 5.24971666641931078667979733510, 5.87160259812856106072880611378, 6.02171262674421941386979504299, 6.39549452819894005964596905258, 6.52114040423523938238352679120, 6.84595645193134669203456303169, 6.93349951101395621475920838167

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