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Properties

Label 8-8025e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.147\times 10^{15}$
Sign $1$
Analytic cond. $1.68612\times 10^{7}$
Root an. cond. $8.00499$
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  + 3·2-s − 4·3-s + 2·4-s − 12·6-s + 8·7-s − 4·8-s + 10·9-s − 4·11-s − 8·12-s + 9·13-s + 24·14-s − 10·16-s + 6·17-s + 30·18-s − 7·19-s − 32·21-s − 12·22-s + 16·23-s + 16·24-s + 27·26-s − 20·27-s + 16·28-s + 29-s − 6·31-s − 9·32-s + 16·33-s + 18·34-s + ⋯
L(s)  = 1  + 2.12·2-s − 2.30·3-s + 4-s − 4.89·6-s + 3.02·7-s − 1.41·8-s + 10/3·9-s − 1.20·11-s − 2.30·12-s + 2.49·13-s + 6.41·14-s − 5/2·16-s + 1.45·17-s + 7.07·18-s − 1.60·19-s − 6.98·21-s − 2.55·22-s + 3.33·23-s + 3.26·24-s + 5.29·26-s − 3.84·27-s + 3.02·28-s + 0.185·29-s − 1.07·31-s − 1.59·32-s + 2.78·33-s + 3.08·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 107^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 107^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 107^{4}\)
Sign: $1$
Analytic conductor: \(1.68612\times 10^{7}\)
Root analytic conductor: \(8.00499\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 107^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(23.91785914\)
\(L(\frac12)\) \(\approx\) \(23.91785914\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
107$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - 3 T + 7 T^{2} - 11 T^{3} + 17 T^{4} - 11 p T^{5} + 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 8 T + 45 T^{2} - 24 p T^{3} + 513 T^{4} - 24 p^{2} T^{5} + 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4 T + 46 T^{2} + 127 T^{3} + 767 T^{4} + 127 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 9 T + 70 T^{2} - 329 T^{3} + 1423 T^{4} - 329 p T^{5} + 70 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 6 T + 47 T^{2} - 200 T^{3} + 1113 T^{4} - 200 p T^{5} + 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 7 T + 66 T^{2} + p^{2} T^{3} + 93 p T^{4} + p^{3} T^{5} + 66 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 16 T + 182 T^{2} - 1313 T^{3} + 7485 T^{4} - 1313 p T^{5} + 182 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - T + 37 T^{2} + 2 T^{3} + 683 T^{4} + 2 p T^{5} + 37 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 6 T + 98 T^{2} + 487 T^{3} + 4207 T^{4} + 487 p T^{5} + 98 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 8 T + 94 T^{2} - 816 T^{3} + 4639 T^{4} - 816 p T^{5} + 94 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 13 T + 189 T^{2} - 1538 T^{3} + 11995 T^{4} - 1538 p T^{5} + 189 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 6 T + 134 T^{2} - 637 T^{3} + 7795 T^{4} - 637 p T^{5} + 134 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 5 T + 112 T^{2} - 767 T^{3} + 6191 T^{4} - 767 p T^{5} + 112 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 5 T + 3 T^{2} + 128 T^{3} + 11 p T^{4} + 128 p T^{5} + 3 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 11 T + 198 T^{2} + 1603 T^{3} + 17389 T^{4} + 1603 p T^{5} + 198 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 6 T + 106 T^{2} + 137 T^{3} + 2947 T^{4} + 137 p T^{5} + 106 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 50 T + 1178 T^{2} - 17099 T^{3} + 167983 T^{4} - 17099 p T^{5} + 1178 p^{2} T^{6} - 50 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 7 T + 194 T^{2} - 1325 T^{3} + 17463 T^{4} - 1325 p T^{5} + 194 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - T + 3 p T^{2} - 78 T^{3} + 21995 T^{4} - 78 p T^{5} + 3 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - T + 4 T^{2} - 27 T^{3} + 6211 T^{4} - 27 p T^{5} + 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 9 T + 323 T^{2} - 2106 T^{3} + 39921 T^{4} - 2106 p T^{5} + 323 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 10 T + 208 T^{2} + 1585 T^{3} + 25085 T^{4} + 1585 p T^{5} + 208 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 23 T + 422 T^{2} - 5705 T^{3} + 64969 T^{4} - 5705 p T^{5} + 422 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.44604697205432906951474763954, −5.30654184492032880237269429684, −5.16058644903724111980937563564, −4.92488314335744388812617618116, −4.84697641549284650678226796121, −4.54617463853556817346407992838, −4.50326972642553439010268111776, −4.39241605831625806207335078905, −4.29103751328791901704720517789, −3.82498496971224604523881846701, −3.71128514774569970687111687683, −3.65126040157047873638412237154, −3.60761337964187914040554143251, −3.05008137151299728388262905702, −2.82305294362512303301115505162, −2.74857070458411026011924469624, −2.23631518389206237928413055390, −2.13413088690365712705368854933, −1.90547283647648496364585658740, −1.59606425910783968269463515152, −1.35476811773186544853354050974, −0.982823163461483582852716627934, −0.806984321038661943352305187997, −0.63364280307848728191314761780, −0.60407783172681225277799708625, 0.60407783172681225277799708625, 0.63364280307848728191314761780, 0.806984321038661943352305187997, 0.982823163461483582852716627934, 1.35476811773186544853354050974, 1.59606425910783968269463515152, 1.90547283647648496364585658740, 2.13413088690365712705368854933, 2.23631518389206237928413055390, 2.74857070458411026011924469624, 2.82305294362512303301115505162, 3.05008137151299728388262905702, 3.60761337964187914040554143251, 3.65126040157047873638412237154, 3.71128514774569970687111687683, 3.82498496971224604523881846701, 4.29103751328791901704720517789, 4.39241605831625806207335078905, 4.50326972642553439010268111776, 4.54617463853556817346407992838, 4.84697641549284650678226796121, 4.92488314335744388812617618116, 5.16058644903724111980937563564, 5.30654184492032880237269429684, 5.44604697205432906951474763954

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