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Properties

Label 8-63e8-1.1-c1e4-0-2
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

−6.51609084366511489436114905005, −6.27942652040284916381657993247, −5.97082052995654945288800583142, −5.63877790699308779216416176248, −5.49137624645987082211133617338, −5.34698656975078951580029466777, −5.09934023807690562615202450444, −5.06933210870832774790793209706, −4.73965146650525008181621090697, −4.46226074337265436859034769152, −4.34817785082819909578099557948, −4.25994559912575286909155259427, −4.21146971700240751097688525848, −3.64201312524224892338881372394, −3.56568903663609783365364805669, −3.49728767493696592114246122214, −2.80134220898881753055743194445, −2.75123493117753039810407004854, −2.58207872135163002831511192968, −2.39717914282193320310745306059, −2.18852339426071782634577451241, −1.99345150355621598533529334680, −1.35099397022673191617344145123, −1.20330645452084263599432185205, −1.05396365634755872690760620279, 0, 0, 0, 0, 1.05396365634755872690760620279, 1.20330645452084263599432185205, 1.35099397022673191617344145123, 1.99345150355621598533529334680, 2.18852339426071782634577451241, 2.39717914282193320310745306059, 2.58207872135163002831511192968, 2.75123493117753039810407004854, 2.80134220898881753055743194445, 3.49728767493696592114246122214, 3.56568903663609783365364805669, 3.64201312524224892338881372394, 4.21146971700240751097688525848, 4.25994559912575286909155259427, 4.34817785082819909578099557948, 4.46226074337265436859034769152, 4.73965146650525008181621090697, 5.06933210870832774790793209706, 5.09934023807690562615202450444, 5.34698656975078951580029466777, 5.49137624645987082211133617338, 5.63877790699308779216416176248, 5.97082052995654945288800583142, 6.27942652040284916381657993247, 6.51609084366511489436114905005

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