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Properties

Label 8-77e8-1.1-c1e4-0-9
Degree $8$
Conductor $1.236\times 10^{15}$
Sign $1$
Analytic cond. $5.02382\times 10^{6}$
Root an. cond. $6.88064$
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  − 2·2-s + 2·3-s + 6·5-s − 4·6-s + 8-s − 7·9-s − 12·10-s + 12·15-s − 16-s − 3·17-s + 14·18-s + 3·19-s − 8·23-s + 2·24-s + 8·25-s − 20·27-s − 3·29-s − 24·30-s + 3·31-s + 10·32-s + 6·34-s − 7·37-s − 6·38-s + 6·40-s + 4·41-s − 8·43-s − 42·45-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 2.68·5-s − 1.63·6-s + 0.353·8-s − 7/3·9-s − 3.79·10-s + 3.09·15-s − 1/4·16-s − 0.727·17-s + 3.29·18-s + 0.688·19-s − 1.66·23-s + 0.408·24-s + 8/5·25-s − 3.84·27-s − 0.557·29-s − 4.38·30-s + 0.538·31-s + 1.76·32-s + 1.02·34-s − 1.15·37-s − 0.973·38-s + 0.948·40-s + 0.624·41-s − 1.21·43-s − 6.26·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(5.02382\times 10^{6}\)
Root analytic conductor: \(6.88064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 7^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad7 \( 1 \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + 7 T^{3} + 13 T^{4} + 7 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$D_{4}$ \( ( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 87 T^{3} + 229 T^{4} - 87 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} - 5 p T^{3} + 153 T^{4} - 5 p^{2} T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 45 T^{2} + 62 T^{3} + 881 T^{4} + 62 p T^{5} + 45 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 47 T^{2} - 36 T^{3} + 919 T^{4} - 36 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 402 p T^{5} + 83 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 62 T^{2} + 9 p T^{3} + 2319 T^{4} + 9 p^{2} T^{5} + 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 100 T^{2} - 299 T^{3} + 4283 T^{4} - 299 p T^{5} + 100 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 767 p T^{5} + 154 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 107 T^{2} - 470 T^{3} + 5491 T^{4} - 470 p T^{5} + 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 162 T^{2} - 1449 T^{3} + 10505 T^{4} - 1449 p T^{5} + 162 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 108 T^{2} + 725 T^{3} + 4961 T^{4} + 725 p T^{5} + 108 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 25 T + 428 T^{2} - 4725 T^{3} + 42353 T^{4} - 4725 p T^{5} + 428 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 238 T^{2} + 2447 T^{3} + 20599 T^{4} + 2447 p T^{5} + 238 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 201 T^{2} + 812 T^{3} + 17469 T^{4} + 812 p T^{5} + 201 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 176 T^{2} + 1649 T^{3} + 20013 T^{4} + 1649 p T^{5} + 176 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 308 T^{2} + 1735 T^{3} + 35911 T^{4} + 1735 p T^{5} + 308 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + T + 96 T^{2} + 1149 T^{3} + 4403 T^{4} + 1149 p T^{5} + 96 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 3419 p T^{5} + 312 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 278 T^{2} - 2415 T^{3} + 30889 T^{4} - 2415 p T^{5} + 278 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.05798499805972919938161130191, −5.98892201783789711222998979549, −5.64280356197647817933561239201, −5.63846754823914894337422823065, −5.54582993106243079003999335572, −5.11003180793352681687570304909, −5.00504155549582620888964497391, −4.85673114870947377295444504781, −4.77970433267581975263668454193, −4.17083793708394515697884430189, −4.06513166189990720440908647266, −3.87258426841584348940132049219, −3.72457194786103346288113937539, −3.44460976843999969877176679586, −3.32704096255109637775555627800, −2.77312062167419797996171170113, −2.69192099102654646400468020231, −2.53869750411443259511435191685, −2.53228367236093518788572954993, −2.21249026627461846506968894763, −2.20063523926006758866616544433, −1.65933045937470579987646583180, −1.50192962034878921344735305803, −1.18952446962072776868560360303, −1.11330511899805609954755746804, 0, 0, 0, 0, 1.11330511899805609954755746804, 1.18952446962072776868560360303, 1.50192962034878921344735305803, 1.65933045937470579987646583180, 2.20063523926006758866616544433, 2.21249026627461846506968894763, 2.53228367236093518788572954993, 2.53869750411443259511435191685, 2.69192099102654646400468020231, 2.77312062167419797996171170113, 3.32704096255109637775555627800, 3.44460976843999969877176679586, 3.72457194786103346288113937539, 3.87258426841584348940132049219, 4.06513166189990720440908647266, 4.17083793708394515697884430189, 4.77970433267581975263668454193, 4.85673114870947377295444504781, 5.00504155549582620888964497391, 5.11003180793352681687570304909, 5.54582993106243079003999335572, 5.63846754823914894337422823065, 5.64280356197647817933561239201, 5.98892201783789711222998979549, 6.05798499805972919938161130191

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