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Properties

Label 8-63e8-1.1-c1e4-0-7
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.00886\times 10^{6}$
Root an. cond. $5.62962$
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-s − 4-s + 2·5-s − 3·8-s + 2·10-s + 5·11-s − 5·13-s − 4·16-s + 6·17-s − 8·19-s − 2·20-s + 5·22-s − 12·23-s − 4·25-s − 5·26-s − 10·29-s − 18·31-s − 32-s + 6·34-s − 8·38-s − 6·40-s − 5·41-s − 7·43-s − 5·44-s − 12·46-s − 21·47-s − 4·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s + 1.50·11-s − 1.38·13-s − 16-s + 1.45·17-s − 1.83·19-s − 0.447·20-s + 1.06·22-s − 2.50·23-s − 4/5·25-s − 0.980·26-s − 1.85·29-s − 3.23·31-s − 0.176·32-s + 1.02·34-s − 1.29·38-s − 0.948·40-s − 0.780·41-s − 1.06·43-s − 0.753·44-s − 1.76·46-s − 3.06·47-s − 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.00886\times 10^{6}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - T + p T^{2} + 3 T^{4} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} + 3 T^{3} + 9 T^{4} + 3 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5 T + 20 T^{2} + 9 T^{3} - 51 T^{4} + 9 p T^{5} + 20 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 5 T + 4 p T^{2} + 175 T^{3} + 1007 T^{4} + 175 p T^{5} + 4 p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 6 T + 23 T^{2} + 48 T^{3} - 363 T^{4} + 48 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 8 T + 4 p T^{2} + 409 T^{3} + 2117 T^{4} + 409 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 128 T^{2} + 861 T^{3} + 4839 T^{4} + 861 p T^{5} + 128 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 10 T + 4 p T^{2} + 780 T^{3} + 5109 T^{4} + 780 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 18 T + 7 p T^{2} + 1810 T^{3} + 11511 T^{4} + 1810 p T^{5} + 7 p^{3} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 70 T^{2} + p T^{3} + 3393 T^{4} + p^{2} T^{5} + 70 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 5 T + 92 T^{2} + 489 T^{3} + 4623 T^{4} + 489 p T^{5} + 92 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 7 T + 142 T^{2} + 727 T^{3} + 8563 T^{4} + 727 p T^{5} + 142 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 21 T + 341 T^{2} + 3408 T^{3} + 28077 T^{4} + 3408 p T^{5} + 341 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 12 T + 4 p T^{2} + 1803 T^{3} + 16773 T^{4} + 1803 p T^{5} + 4 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 6 T + 128 T^{2} - 861 T^{3} + 8331 T^{4} - 861 p T^{5} + 128 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 20 T + 271 T^{2} + 3010 T^{3} + 26663 T^{4} + 3010 p T^{5} + 271 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 5 T + 196 T^{2} + 931 T^{3} + 17639 T^{4} + 931 p T^{5} + 196 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 9 T + 257 T^{2} + 1782 T^{3} + 26655 T^{4} + 1782 p T^{5} + 257 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 6 T + 142 T^{2} + 19 p T^{3} + 12363 T^{4} + 19 p^{2} T^{5} + 142 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 10 T + 232 T^{2} + 2365 T^{3} + 24181 T^{4} + 2365 p T^{5} + 232 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 9 T + 206 T^{2} - 531 T^{3} + 15315 T^{4} - 531 p T^{5} + 206 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 22 T + 470 T^{2} + 5925 T^{3} + 67797 T^{4} + 5925 p T^{5} + 470 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 9 T + 130 T^{2} + 163 T^{3} + 9279 T^{4} + 163 p T^{5} + 130 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.19139980599586353538077015910, −6.02875108968461969174417035957, −5.95411163239373925081626069351, −5.75046688377008109227870114205, −5.68163287425207702117539735922, −5.43503401162354902989970157032, −5.16150231440569629864531336583, −5.05557301060599780296141997990, −4.77131346516876758841086362358, −4.49028345069840469569321045550, −4.31396004298104478610854872719, −4.15609908251833448234863404732, −4.15354854845866238481336312038, −3.72140152173220701665678628507, −3.49408092049998080400970056895, −3.40472449264272195698221173636, −3.15139344144005702265361638040, −3.08269171545153371394427336035, −2.55727868543143765386821822517, −2.23684433038896207050193428539, −1.99488518978230516022067823977, −1.89599294907052143510114325066, −1.68125911464144767509317796139, −1.40664528018848430030965982015, −1.35125593057213708670583987602, 0, 0, 0, 0, 1.35125593057213708670583987602, 1.40664528018848430030965982015, 1.68125911464144767509317796139, 1.89599294907052143510114325066, 1.99488518978230516022067823977, 2.23684433038896207050193428539, 2.55727868543143765386821822517, 3.08269171545153371394427336035, 3.15139344144005702265361638040, 3.40472449264272195698221173636, 3.49408092049998080400970056895, 3.72140152173220701665678628507, 4.15354854845866238481336312038, 4.15609908251833448234863404732, 4.31396004298104478610854872719, 4.49028345069840469569321045550, 4.77131346516876758841086362358, 5.05557301060599780296141997990, 5.16150231440569629864531336583, 5.43503401162354902989970157032, 5.68163287425207702117539735922, 5.75046688377008109227870114205, 5.95411163239373925081626069351, 6.02875108968461969174417035957, 6.19139980599586353538077015910

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