Properties

Label 8-45e8-1.1-c1e4-0-7
Degree $8$
Conductor $1.682\times 10^{13}$
Sign $1$
Analytic cond. $68361.0$
Root an. cond. $4.02115$
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-s + 6·11-s + 16-s + 10·19-s + 14·31-s − 6·41-s − 6·44-s − 4·49-s − 18·59-s + 26·61-s − 5·64-s − 18·71-s − 10·76-s + 4·79-s + 12·89-s + 54·101-s + 16·109-s − 5·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.80·11-s + 1/4·16-s + 2.29·19-s + 2.51·31-s − 0.937·41-s − 0.904·44-s − 4/7·49-s − 2.34·59-s + 3.32·61-s − 5/8·64-s − 2.13·71-s − 1.14·76-s + 0.450·79-s + 1.27·89-s + 5.37·101-s + 1.53·109-s − 0.454·121-s − 1.25·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(68361.0\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.228233149\)
\(L(\frac12)\) \(\approx\) \(6.228233149\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good2$D_4$ \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + T^{2} + 264 T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 61 T^{2} + 1500 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 64 T^{2} + 1950 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 85 T^{2} + 3876 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 112 T^{2} + 6366 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 40 T^{2} + 2718 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 64 T^{2} + 8814 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 229 T^{2} + 23100 T^{4} + 229 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 160 T^{2} + 16878 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 + 40 T^{2} - 10482 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.59738334229065639048379017336, −6.27246707747713533034509224587, −6.06572854375836656557161130859, −6.02139997965520888865495032750, −5.60069385889621025499205089528, −5.50621178738590335458374920171, −5.07862485814308290615534766321, −4.96744227552950705213470589408, −4.92871129816834904903256380569, −4.50885840635276226717378279135, −4.31692788287165865990642589343, −4.26741208825602059776428068777, −3.87060206227427494683870901040, −3.48137038560412142421624022065, −3.38729919801291824966886157973, −3.37536044207277499105533566234, −2.97628184686321508236170717143, −2.58165870243333611047692684736, −2.58132588029700592586034894046, −1.89093724535037867983692172051, −1.71162841224541497355043549549, −1.50329982345674225479763102755, −1.04717586211223452855631539167, −0.70574984178323114281837497981, −0.58983640741714363049922167969, 0.58983640741714363049922167969, 0.70574984178323114281837497981, 1.04717586211223452855631539167, 1.50329982345674225479763102755, 1.71162841224541497355043549549, 1.89093724535037867983692172051, 2.58132588029700592586034894046, 2.58165870243333611047692684736, 2.97628184686321508236170717143, 3.37536044207277499105533566234, 3.38729919801291824966886157973, 3.48137038560412142421624022065, 3.87060206227427494683870901040, 4.26741208825602059776428068777, 4.31692788287165865990642589343, 4.50885840635276226717378279135, 4.92871129816834904903256380569, 4.96744227552950705213470589408, 5.07862485814308290615534766321, 5.50621178738590335458374920171, 5.60069385889621025499205089528, 6.02139997965520888865495032750, 6.06572854375836656557161130859, 6.27246707747713533034509224587, 6.59738334229065639048379017336

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