Properties

Label 8-31e8-1.1-c1e4-0-9
Degree $8$
Conductor $852891037441$
Sign $1$
Analytic cond. $3467.38$
Root an. cond. $2.77013$
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  + 2·2-s − 2·3-s − 3·4-s + 3·5-s − 4·6-s − 6·7-s − 10·8-s + 7·9-s + 6·10-s − 6·11-s + 6·12-s + 3·13-s − 12·14-s − 6·15-s − 4·17-s + 14·18-s − 10·19-s − 9·20-s + 12·21-s − 12·22-s + 4·23-s + 20·24-s + 11·25-s + 6·26-s − 22·27-s + 18·28-s + 30·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s − 3/2·4-s + 1.34·5-s − 1.63·6-s − 2.26·7-s − 3.53·8-s + 7/3·9-s + 1.89·10-s − 1.80·11-s + 1.73·12-s + 0.832·13-s − 3.20·14-s − 1.54·15-s − 0.970·17-s + 3.29·18-s − 2.29·19-s − 2.01·20-s + 2.61·21-s − 2.55·22-s + 0.834·23-s + 4.08·24-s + 11/5·25-s + 1.17·26-s − 4.23·27-s + 3.40·28-s + 5.57·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31^{8}\)
Sign: $1$
Analytic conductor: \(3467.38\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7917420318\)
\(L(\frac12)\) \(\approx\) \(0.7917420318\)
\(L(\frac{3}{2})\) 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)$
bad31 \( 1 \)
good2$D_{4}$ \( ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
3$C_2^2$ \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_4\times C_2$ \( 1 + 6 T + 10 T^{2} + 24 T^{3} + 159 T^{4} + 24 p T^{5} + 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 3 T - 8 T^{2} + 27 T^{3} + 3 T^{4} + 27 p T^{5} - 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 4 T - p T^{2} - 4 T^{3} + 528 T^{4} - 4 p T^{5} - p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 4 T - 57 T^{2} - 4 T^{3} + 3368 T^{4} - 4 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 4 T - 50 T^{2} - 64 T^{3} + 2019 T^{4} - 64 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 7 T - 48 T^{2} + 77 T^{3} + 5453 T^{4} + 77 p T^{5} - 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 12 T + 47 T^{2} - 108 T^{3} - 1032 T^{4} - 108 p T^{5} + 47 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 10 T - 23 T^{2} + 50 T^{3} + 5748 T^{4} + 50 p T^{5} - 23 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 4 T - 117 T^{2} + 4 T^{3} + 12128 T^{4} + 4 p T^{5} - 117 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 11 T - 20 T^{2} + 11 T^{3} + 6249 T^{4} + 11 p T^{5} - 20 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 3 T - 38 T^{2} + 297 T^{3} - 3777 T^{4} + 297 p T^{5} - 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 3 T - 128 T^{2} + 87 T^{3} + 11133 T^{4} + 87 p T^{5} - 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 15 T + 233 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 24 T + 293 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.90403984327971351532389130364, −6.88271757958898352983330730347, −6.39751679491253939192114953790, −6.31364475381800771740623459203, −6.28992385563979497663963529521, −6.15986399964514616504511065712, −5.95097819380783799009366551168, −5.21810411800881077048588835710, −5.19845117076776146872257342730, −5.18183032494642023202372265547, −4.93285881413882121508174583192, −4.51341724400154328630822860052, −4.45474619958065899136456612759, −4.30972077593198420737730185885, −4.25753568610799782494438246153, −3.43428919097978450756725829672, −3.28028795792254990178047517501, −3.26764808434056377771295193647, −3.09638441642875175622430376000, −2.41800467465297864579022774625, −2.19086084405425483947862188507, −1.81520953745442871453278216255, −1.23509293784862175622206171081, −0.61312467116054062893572141416, −0.30269298151513855362829477622, 0.30269298151513855362829477622, 0.61312467116054062893572141416, 1.23509293784862175622206171081, 1.81520953745442871453278216255, 2.19086084405425483947862188507, 2.41800467465297864579022774625, 3.09638441642875175622430376000, 3.26764808434056377771295193647, 3.28028795792254990178047517501, 3.43428919097978450756725829672, 4.25753568610799782494438246153, 4.30972077593198420737730185885, 4.45474619958065899136456612759, 4.51341724400154328630822860052, 4.93285881413882121508174583192, 5.18183032494642023202372265547, 5.19845117076776146872257342730, 5.21810411800881077048588835710, 5.95097819380783799009366551168, 6.15986399964514616504511065712, 6.28992385563979497663963529521, 6.31364475381800771740623459203, 6.39751679491253939192114953790, 6.88271757958898352983330730347, 6.90403984327971351532389130364

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