Properties

Label 8-1568e4-1.1-c1e4-0-13
Degree $8$
Conductor $6.045\times 10^{12}$
Sign $1$
Analytic cond. $24574.9$
Root an. cond. $3.53843$
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  + 4·9-s − 4·11-s − 16·23-s + 10·25-s + 24·29-s + 4·37-s + 24·43-s − 12·53-s − 24·67-s − 16·71-s − 24·79-s + 9·81-s − 16·99-s − 8·107-s − 36·109-s − 48·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + ⋯
L(s)  = 1  + 4/3·9-s − 1.20·11-s − 3.33·23-s + 2·25-s + 4.45·29-s + 0.657·37-s + 3.65·43-s − 1.64·53-s − 2.93·67-s − 1.89·71-s − 2.70·79-s + 81-s − 1.60·99-s − 0.773·107-s − 3.44·109-s − 4.51·113-s + 2.36·121-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 − 2.76·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(24574.9\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.789291815\)
\(L(\frac12)\) \(\approx\) \(3.789291815\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^3$ \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 20 T^{2} + 39 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 + 10 T^{2} - 861 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 44 T^{2} - 1545 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 90 T^{2} + 4379 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 160 T^{2} + 17679 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.74938163587396926776535545588, −6.33790430444952708972451610287, −6.28569112219255007090849611763, −6.22918118993106263830181221916, −5.90925197614927873912122679531, −5.66098856986975973444086923494, −5.48991239288144907998746428731, −5.19084452289736276253001716497, −4.85581111930457391059818231875, −4.54460729804461556703721270043, −4.47696979852642514210404130513, −4.42840276094536100430373546655, −4.15141449911643821681643213272, −3.96305907775527282941295261681, −3.64608692824206332339847020434, −3.00064725385592036977044219661, −2.86029502199837195853982099094, −2.81853297801407290945965453442, −2.65008365788140408464944008685, −2.35622853945817934315389231381, −1.65487196298940704089874676797, −1.53054612525723678399969761225, −1.37486385039568386140944207989, −0.68713370277032523681994294254, −0.46977527012987877848357091576, 0.46977527012987877848357091576, 0.68713370277032523681994294254, 1.37486385039568386140944207989, 1.53054612525723678399969761225, 1.65487196298940704089874676797, 2.35622853945817934315389231381, 2.65008365788140408464944008685, 2.81853297801407290945965453442, 2.86029502199837195853982099094, 3.00064725385592036977044219661, 3.64608692824206332339847020434, 3.96305907775527282941295261681, 4.15141449911643821681643213272, 4.42840276094536100430373546655, 4.47696979852642514210404130513, 4.54460729804461556703721270043, 4.85581111930457391059818231875, 5.19084452289736276253001716497, 5.48991239288144907998746428731, 5.66098856986975973444086923494, 5.90925197614927873912122679531, 6.22918118993106263830181221916, 6.28569112219255007090849611763, 6.33790430444952708972451610287, 6.74938163587396926776535545588

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