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Properties

Label 8-55e8-1.1-c1e4-0-4
Degree $8$
Conductor $8.373\times 10^{13}$
Sign $1$
Analytic cond. $340415.$
Root an. cond. $4.91474$
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  − 3·2-s + 2·3-s + 4·4-s − 6·6-s − 11·7-s − 4·8-s − 4·9-s + 8·12-s − 7·13-s + 33·14-s + 2·16-s − 3·17-s + 12·18-s + 12·19-s − 22·21-s + 9·23-s − 8·24-s + 21·26-s − 11·27-s − 44·28-s − 8·29-s + 3·31-s + 5·32-s + 9·34-s − 16·36-s + 3·37-s − 36·38-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 2·4-s − 2.44·6-s − 4.15·7-s − 1.41·8-s − 4/3·9-s + 2.30·12-s − 1.94·13-s + 8.81·14-s + 1/2·16-s − 0.727·17-s + 2.82·18-s + 2.75·19-s − 4.80·21-s + 1.87·23-s − 1.63·24-s + 4.11·26-s − 2.11·27-s − 8.31·28-s − 1.48·29-s + 0.538·31-s + 0.883·32-s + 1.54·34-s − 8/3·36-s + 0.493·37-s − 5.83·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(340415.\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
11 \( 1 \)
good2$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 11 T^{4} + 7 p T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 8 T^{2} - 13 T^{3} + 35 T^{4} - 13 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 67 T^{2} + 276 T^{3} + 845 T^{4} + 276 p T^{5} + 67 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 59 T^{2} + 264 T^{3} + 1185 T^{4} + 264 p T^{5} + 59 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 60 T^{2} + 127 T^{3} + 1451 T^{4} + 127 p T^{5} + 60 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 122 T^{2} - 749 T^{3} + 3939 T^{4} - 749 p T^{5} + 122 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 38 T^{2} + 85 T^{3} - 979 T^{4} + 85 p T^{5} + 38 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 112 T^{2} + 601 T^{3} + 4759 T^{4} + 601 p T^{5} + 112 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 276 T^{3} + 3945 T^{4} - 276 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 92 T^{2} - 305 T^{3} + 4221 T^{4} - 305 p T^{5} + 92 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 118 T^{2} + 479 T^{3} + 5815 T^{4} + 479 p T^{5} + 118 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 293 T^{2} + 2900 T^{3} + 21441 T^{4} + 2900 p T^{5} + 293 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 137 T^{2} - 290 T^{3} + 8531 T^{4} - 290 p T^{5} + 137 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 245 T^{2} - 1776 T^{3} + 20351 T^{4} - 1776 p T^{5} + 245 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 127 T^{2} + 44 T^{3} + 4999 T^{4} + 44 p T^{5} + 127 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 203 T^{2} - 642 T^{3} + 17379 T^{4} - 642 p T^{5} + 203 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - T + 186 T^{2} + 37 T^{3} + 15845 T^{4} + 37 p T^{5} + 186 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 186 T^{2} + 925 T^{3} + 8531 T^{4} + 925 p T^{5} + 186 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 218 T^{2} - 1375 T^{3} + 20781 T^{4} - 1375 p T^{5} + 218 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 220 T^{2} - 487 T^{3} + 20123 T^{4} - 487 p T^{5} + 220 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 375 T^{2} + 3710 T^{3} + 48443 T^{4} + 3710 p T^{5} + 375 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 206 T^{2} - 400 T^{3} + 21551 T^{4} - 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 332 T^{2} + 1836 T^{3} + 45565 T^{4} + 1836 p T^{5} + 332 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.64034510553072870855756965958, −6.53158591624653297168594097835, −6.43513748409708196044495868951, −6.07450536497259733844390902359, −5.90481777461327683797876963225, −5.58118179025102737589443002155, −5.30317004373139834678407365927, −5.29182114688552364639002774251, −5.27172735264673565384795411215, −4.70928733263398511268318121431, −4.60115969118507816059526945193, −4.16594000799710546674948634642, −3.95497674601835551487245189493, −3.46896006353525058391645238817, −3.46193622023666107912745508993, −3.13828672957115936763113209835, −3.09925626818282474702696627526, −2.98578651915386719364933636422, −2.92360927880467552357364148719, −2.37484496001582360394807968949, −2.36792514318448067927864423377, −2.17516992994483788831485047740, −1.51068037498304712454233728149, −1.06077041459891431252505156392, −0.967911907256866605583460300236, 0, 0, 0, 0, 0.967911907256866605583460300236, 1.06077041459891431252505156392, 1.51068037498304712454233728149, 2.17516992994483788831485047740, 2.36792514318448067927864423377, 2.37484496001582360394807968949, 2.92360927880467552357364148719, 2.98578651915386719364933636422, 3.09925626818282474702696627526, 3.13828672957115936763113209835, 3.46193622023666107912745508993, 3.46896006353525058391645238817, 3.95497674601835551487245189493, 4.16594000799710546674948634642, 4.60115969118507816059526945193, 4.70928733263398511268318121431, 5.27172735264673565384795411215, 5.29182114688552364639002774251, 5.30317004373139834678407365927, 5.58118179025102737589443002155, 5.90481777461327683797876963225, 6.07450536497259733844390902359, 6.43513748409708196044495868951, 6.53158591624653297168594097835, 6.64034510553072870855756965958

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