Properties

Label 8-70e8-1.1-c1e4-0-3
Degree $8$
Conductor $5.765\times 10^{14}$
Sign $1$
Analytic cond. $2.34364\times 10^{6}$
Root an. cond. $6.25513$
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  − 6·9-s − 4·11-s + 4·23-s − 8·29-s + 8·37-s + 24·43-s + 20·53-s − 20·67-s + 40·71-s − 28·79-s + 14·81-s + 24·99-s + 32·107-s − 12·109-s + 64·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 173-s + ⋯
L(s)  = 1  − 2·9-s − 1.20·11-s + 0.834·23-s − 1.48·29-s + 1.31·37-s + 3.65·43-s + 2.74·53-s − 2.44·67-s + 4.74·71-s − 3.15·79-s + 14/9·81-s + 2.41·99-s + 3.09·107-s − 1.14·109-s + 6.02·113-s + 6/11·121-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 − 3.53·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.34364\times 10^{6}\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.228081850\)
\(L(\frac12)\) \(\approx\) \(3.228081850\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 2 p T^{2} + 22 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 44 T^{2} + 982 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 40 T^{2} + 802 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 54 T^{2} + 1526 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 134 T^{2} + 7726 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 12 T + 117 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 104 T^{2} + 6802 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 206 T^{2} + 17446 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 26 T^{2} + 1486 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 20 T + 197 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 22 T^{2} + 4654 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 102 T^{2} + 15254 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 86 T^{2} + 7566 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 26 T^{2} + 15342 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.93103339035511124338737655038, −5.45216282915463475216331133690, −5.43718188702745864673866994585, −5.39558547170163201937129346998, −5.37978938935249853151622372707, −4.70567118829036949298036944979, −4.62822159024285708917792211729, −4.46672060887746376130340083553, −4.45239647435002375701903046815, −4.01660210341536288623796236494, −3.73805032795989197811875815239, −3.64319963797337231779417877022, −3.49340572156087174183561772818, −2.96573568440868785956777572799, −2.89969950553206809920443604195, −2.79127988309727595121317371592, −2.72553624644616607712510073607, −2.29512599323517187447256428241, −2.03109353934158492908520839028, −1.87927474761173300034679782465, −1.78964527129714277750024450362, −0.892667400537121594217391588210, −0.808146247452472961744912424162, −0.70399133812821928059399076447, −0.31124765500547730281831446305, 0.31124765500547730281831446305, 0.70399133812821928059399076447, 0.808146247452472961744912424162, 0.892667400537121594217391588210, 1.78964527129714277750024450362, 1.87927474761173300034679782465, 2.03109353934158492908520839028, 2.29512599323517187447256428241, 2.72553624644616607712510073607, 2.79127988309727595121317371592, 2.89969950553206809920443604195, 2.96573568440868785956777572799, 3.49340572156087174183561772818, 3.64319963797337231779417877022, 3.73805032795989197811875815239, 4.01660210341536288623796236494, 4.45239647435002375701903046815, 4.46672060887746376130340083553, 4.62822159024285708917792211729, 4.70567118829036949298036944979, 5.37978938935249853151622372707, 5.39558547170163201937129346998, 5.43718188702745864673866994585, 5.45216282915463475216331133690, 5.93103339035511124338737655038

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