Properties

Label 8-280e4-1.1-c1e4-0-9
Degree $8$
Conductor $6146560000$
Sign $1$
Analytic cond. $24.9885$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 6·3-s + 8·4-s − 2·5-s + 24·6-s − 8·8-s + 15·9-s + 8·10-s − 4·11-s − 48·12-s − 12·13-s + 12·15-s − 4·16-s + 6·17-s − 60·18-s − 18·19-s − 16·20-s + 16·22-s − 24·23-s + 48·24-s + 25-s + 48·26-s − 18·27-s − 48·30-s + 6·31-s + 32·32-s + 24·33-s + ⋯
L(s)  = 1  − 2.82·2-s − 3.46·3-s + 4·4-s − 0.894·5-s + 9.79·6-s − 2.82·8-s + 5·9-s + 2.52·10-s − 1.20·11-s − 13.8·12-s − 3.32·13-s + 3.09·15-s − 16-s + 1.45·17-s − 14.1·18-s − 4.12·19-s − 3.57·20-s + 3.41·22-s − 5.00·23-s + 9.79·24-s + 1/5·25-s + 9.41·26-s − 3.46·27-s − 8.76·30-s + 1.07·31-s + 5.65·32-s + 4.17·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(24.9885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
11$D_4\times C_2$ \( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 40 T^{2} - 168 T^{3} + 699 T^{4} - 168 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 18 T + 164 T^{2} + 1008 T^{3} + 4827 T^{4} + 1008 p T^{5} + 164 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 24 T + 285 T^{2} + 2232 T^{3} + 12536 T^{4} + 2232 p T^{5} + 285 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 6 T^{2} - 661 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 6 T - 8 T^{2} + 108 T^{3} - 141 T^{4} + 108 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 12 T + 26 T^{2} + 288 T^{3} + 4947 T^{4} + 288 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 18 T + 216 T^{2} + 1944 T^{3} + 14579 T^{4} + 1944 p T^{5} + 216 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T + 124 T^{2} - 672 T^{3} + 9771 T^{4} - 672 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T - 11 T^{2} + 396 T^{3} + 11520 T^{4} + 396 p T^{5} - 11 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 2 T - 23 T^{2} - 214 T^{3} - 4028 T^{4} - 214 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 180 T^{2} + 17594 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 30 T + 512 T^{2} + 6360 T^{3} + 61515 T^{4} + 6360 p T^{5} + 512 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 6 T + 164 T^{2} - 912 T^{3} + 17811 T^{4} - 912 p T^{5} + 164 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 110 T^{2} + 6003 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 30 T + 517 T^{2} - 6510 T^{3} + 65868 T^{4} - 6510 p T^{5} + 517 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \)
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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

−9.354783385882457108404527295431, −8.614098690006061323837314799657, −8.555486439842126471054043189548, −8.417110619017076391297615659500, −8.077371125147485741836240099910, −7.893780494438213361211865683266, −7.57393607370788494023355545937, −7.49070242268475714322864488203, −7.46974911779765507208569119625, −6.66098103290827440423631800368, −6.40660556810252431110409381198, −6.37421838365332261072671762417, −6.27828555228829330937765384916, −5.76425904254147146425575020768, −5.76279013121502496092617392208, −4.85780221458947423922527835508, −4.85079335238427475029557264149, −4.82628215577059390618285864553, −4.57650819351118637474624481275, −4.06518277522645112543534845297, −3.60869616102761753444986059970, −2.77494325968640988911933112943, −2.29000768884661499444346621685, −1.98276865087428115138434858849, −1.68517863689454721011176667104, 0, 0, 0, 0, 1.68517863689454721011176667104, 1.98276865087428115138434858849, 2.29000768884661499444346621685, 2.77494325968640988911933112943, 3.60869616102761753444986059970, 4.06518277522645112543534845297, 4.57650819351118637474624481275, 4.82628215577059390618285864553, 4.85079335238427475029557264149, 4.85780221458947423922527835508, 5.76279013121502496092617392208, 5.76425904254147146425575020768, 6.27828555228829330937765384916, 6.37421838365332261072671762417, 6.40660556810252431110409381198, 6.66098103290827440423631800368, 7.46974911779765507208569119625, 7.49070242268475714322864488203, 7.57393607370788494023355545937, 7.893780494438213361211865683266, 8.077371125147485741836240099910, 8.417110619017076391297615659500, 8.555486439842126471054043189548, 8.614098690006061323837314799657, 9.354783385882457108404527295431

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