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Properties

Label 8-605e4-1.1-c1e4-0-9
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 2·3-s + 4·4-s + 4·5-s − 6·6-s + 11·7-s + 4·8-s − 4·9-s + 12·10-s − 8·12-s + 7·13-s + 33·14-s − 8·15-s + 2·16-s + 3·17-s − 12·18-s + 12·19-s + 16·20-s − 22·21-s − 9·23-s − 8·24-s + 10·25-s + 21·26-s + 11·27-s + 44·28-s − 8·29-s − 24·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.15·3-s + 2·4-s + 1.78·5-s − 2.44·6-s + 4.15·7-s + 1.41·8-s − 4/3·9-s + 3.79·10-s − 2.30·12-s + 1.94·13-s + 8.81·14-s − 2.06·15-s + 1/2·16-s + 0.727·17-s − 2.82·18-s + 2.75·19-s + 3.57·20-s − 4.80·21-s − 1.87·23-s − 1.63·24-s + 2·25-s + 4.11·26-s + 2.11·27-s + 8.31·28-s − 1.48·29-s − 4.38·30-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.30570261\)
\(L(\frac12)\) \(\approx\) \(16.30570261\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
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

−7.60935323589162439937982439355, −7.44807360774094599999828725945, −7.37948948187619310636626378580, −6.59981833869442945804463241546, −6.58529880887731459114525705933, −6.10107983838590632936608903311, −5.91438795048445607648537405342, −5.82211887530809714242873864901, −5.75307391154236322311450590435, −5.33322061232929930404373761811, −5.17594273225296416952173819251, −5.08869954135693275926803764032, −5.05390119079528486129013001736, −4.56782242226678921392832011731, −4.30527704672202159909961329509, −4.07944597503032509139738913172, −3.86494629608583165998401050357, −3.19930343251278315123318221028, −3.13269866437963588658453419520, −2.82302347031775843151032153474, −2.08303021296616042965844033710, −1.95688892379934227145845556046, −1.58891951458805395883110651479, −1.41060291378352900714376948884, −0.948431305586601174813472817241, 0.948431305586601174813472817241, 1.41060291378352900714376948884, 1.58891951458805395883110651479, 1.95688892379934227145845556046, 2.08303021296616042965844033710, 2.82302347031775843151032153474, 3.13269866437963588658453419520, 3.19930343251278315123318221028, 3.86494629608583165998401050357, 4.07944597503032509139738913172, 4.30527704672202159909961329509, 4.56782242226678921392832011731, 5.05390119079528486129013001736, 5.08869954135693275926803764032, 5.17594273225296416952173819251, 5.33322061232929930404373761811, 5.75307391154236322311450590435, 5.82211887530809714242873864901, 5.91438795048445607648537405342, 6.10107983838590632936608903311, 6.58529880887731459114525705933, 6.59981833869442945804463241546, 7.37948948187619310636626378580, 7.44807360774094599999828725945, 7.60935323589162439937982439355

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