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Properties

Label 8-8512e4-1.1-c1e4-0-5
Degree $8$
Conductor $5.250\times 10^{15}$
Sign $1$
Analytic cond. $2.13419\times 10^{7}$
Root an. cond. $8.24431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s − 5·9-s + 2·11-s − 6·17-s − 4·19-s + 8·21-s + 4·23-s − 12·25-s − 16·27-s − 2·29-s − 18·31-s + 4·33-s + 16·37-s − 10·41-s − 4·43-s − 4·47-s + 10·49-s − 12·51-s − 10·53-s − 8·57-s − 4·59-s + 16·61-s − 20·63-s − 10·67-s + 8·69-s + 4·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s − 5/3·9-s + 0.603·11-s − 1.45·17-s − 0.917·19-s + 1.74·21-s + 0.834·23-s − 2.39·25-s − 3.07·27-s − 0.371·29-s − 3.23·31-s + 0.696·33-s + 2.63·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 10/7·49-s − 1.68·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s + 2.04·61-s − 2.51·63-s − 1.22·67-s + 0.963·69-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 7^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2.13419\times 10^{7}\)
Root analytic conductor: \(8.24431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 7^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + p^{2} T^{2} - 4 p T^{3} + 35 T^{4} - 4 p^{2} T^{5} + p^{4} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 12 T^{2} + 4 T^{3} + 77 T^{4} + 4 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 41 T^{2} - 64 T^{3} + 661 T^{4} - 64 p T^{5} + 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} - 12 T^{3} + 597 T^{4} - 12 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 55 T^{2} + 240 T^{3} + 1363 T^{4} + 240 p T^{5} + 55 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} - 20 T^{3} + 2329 T^{4} - 20 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 191 T^{2} + 1548 T^{3} + 9967 T^{4} + 1548 p T^{5} + 191 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 214 T^{2} - 1824 T^{3} + 12995 T^{4} - 1824 p T^{5} + 214 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 149 T^{2} + 1044 T^{3} + 9119 T^{4} + 1044 p T^{5} + 149 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 136 T^{2} + 564 T^{3} + 7982 T^{4} + 564 p T^{5} + 136 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 100 T^{2} + 236 T^{3} + 5869 T^{4} + 236 p T^{5} + 100 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 149 T^{2} + 1160 T^{3} + 12025 T^{4} + 1160 p T^{5} + 149 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 164 T^{2} + 276 T^{3} + 11933 T^{4} + 276 p T^{5} + 164 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 268 T^{2} - 2708 T^{3} + 24829 T^{4} - 2708 p T^{5} + 268 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 63 T^{2} + 500 T^{3} + 6785 T^{4} + 500 p T^{5} + 63 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 186 T^{2} - 648 T^{3} + 16883 T^{4} - 648 p T^{5} + 186 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 241 T^{2} + 2728 T^{3} + 28175 T^{4} + 2728 p T^{5} + 241 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 232 T^{2} - 1028 T^{3} + 24574 T^{4} - 1028 p T^{5} + 232 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 143 T^{2} + 444 T^{3} + 2903 T^{4} + 444 p T^{5} + 143 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 324 T^{2} + 2760 T^{3} + 23334 T^{4} + 2760 p T^{5} + 324 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 292 T^{2} + 2196 T^{3} + 28373 T^{4} + 2196 p T^{5} + 292 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.95075545999582350825606007645, −5.48212765585664647992745999755, −5.38099612166610403782668872593, −5.28493770556389679439676733275, −5.28262813211443446544550377372, −4.98234702550663700359205703622, −4.55504931034111227194636656236, −4.54094416207378581137257641034, −4.34405853922864981738932537058, −3.94798416184443200992526932713, −3.90498032103384897440003823015, −3.88267534284319733310169273773, −3.81427374526897454527801807570, −3.13609457594094352079327535666, −3.12459966626566605167462391434, −3.03648104188588296086412550086, −2.86243340239432620345550528426, −2.40334438065332715198733716394, −2.31878075705183782272250273568, −2.05064063720648113313294429442, −1.96061400315172222764464276271, −1.91508048911240360072665558369, −1.25876834028679164557137661540, −1.24348226601184896814241054871, −1.22070961053825409467541695216, 0, 0, 0, 0, 1.22070961053825409467541695216, 1.24348226601184896814241054871, 1.25876834028679164557137661540, 1.91508048911240360072665558369, 1.96061400315172222764464276271, 2.05064063720648113313294429442, 2.31878075705183782272250273568, 2.40334438065332715198733716394, 2.86243340239432620345550528426, 3.03648104188588296086412550086, 3.12459966626566605167462391434, 3.13609457594094352079327535666, 3.81427374526897454527801807570, 3.88267534284319733310169273773, 3.90498032103384897440003823015, 3.94798416184443200992526932713, 4.34405853922864981738932537058, 4.54094416207378581137257641034, 4.55504931034111227194636656236, 4.98234702550663700359205703622, 5.28262813211443446544550377372, 5.28493770556389679439676733275, 5.38099612166610403782668872593, 5.48212765585664647992745999755, 5.95075545999582350825606007645

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