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Properties

Label 8-2016e4-1.1-c3e4-0-2
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $2.00182\times 10^{8}$
Root an. cond. $10.9063$
Motivic weight $3$
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  − 10·5-s − 28·7-s + 22·11-s + 20·13-s + 10·17-s − 20·19-s + 158·23-s − 220·25-s − 124·29-s − 44·31-s + 280·35-s + 268·37-s − 298·41-s + 104·43-s + 364·47-s + 490·49-s − 176·53-s − 220·55-s + 300·59-s + 556·61-s − 200·65-s − 512·67-s + 266·71-s − 320·73-s − 616·77-s + 184·79-s − 728·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 0.603·11-s + 0.426·13-s + 0.142·17-s − 0.241·19-s + 1.43·23-s − 1.75·25-s − 0.794·29-s − 0.254·31-s + 1.35·35-s + 1.19·37-s − 1.13·41-s + 0.368·43-s + 1.12·47-s + 10/7·49-s − 0.456·53-s − 0.539·55-s + 0.661·59-s + 1.16·61-s − 0.381·65-s − 0.933·67-s + 0.444·71-s − 0.513·73-s − 0.911·77-s + 0.262·79-s − 0.962·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2.00182\times 10^{8}\)
Root analytic conductor: \(10.9063\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T + 64 p T^{2} + 3622 T^{3} + 50478 T^{4} + 3622 p^{3} T^{5} + 64 p^{7} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 p T + 2760 T^{2} + 1134 p T^{3} + 2859102 T^{4} + 1134 p^{4} T^{5} + 2760 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 4164 T^{2} - 85724 T^{3} + 9106022 T^{4} - 85724 p^{3} T^{5} + 4164 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 10632 T^{2} - 91102 T^{3} + 75203662 T^{4} - 91102 p^{3} T^{5} + 10632 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 13724 T^{2} + 199060 T^{3} + 114551670 T^{4} + 199060 p^{3} T^{5} + 13724 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 158 T + 50352 T^{2} - 5323734 T^{3} + 922012222 T^{4} - 5323734 p^{3} T^{5} + 50352 p^{6} T^{6} - 158 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 124 T + 61972 T^{2} + 6615172 T^{3} + 2194974870 T^{4} + 6615172 p^{3} T^{5} + 61972 p^{6} T^{6} + 124 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 44 T + 54124 T^{2} + 9682172 T^{3} + 1373933478 T^{4} + 9682172 p^{3} T^{5} + 54124 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 268 T + 169892 T^{2} - 34032292 T^{3} + 12680601926 T^{4} - 34032292 p^{3} T^{5} + 169892 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 298 T + 253368 T^{2} + 54484142 T^{3} + 25744646030 T^{4} + 54484142 p^{3} T^{5} + 253368 p^{6} T^{6} + 298 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 104 T + 288908 T^{2} - 22902568 T^{3} + 33466593270 T^{4} - 22902568 p^{3} T^{5} + 288908 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 364 T + 435532 T^{2} - 109124860 T^{3} + 68532881190 T^{4} - 109124860 p^{3} T^{5} + 435532 p^{6} T^{6} - 364 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 176 T + 289444 T^{2} - 10155056 T^{3} + 39281098070 T^{4} - 10155056 p^{3} T^{5} + 289444 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 300 T + 143868 T^{2} + 27043700 T^{3} - 24108912938 T^{4} + 27043700 p^{3} T^{5} + 143868 p^{6} T^{6} - 300 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 556 T + 784004 T^{2} - 324087844 T^{3} + 262093377062 T^{4} - 324087844 p^{3} T^{5} + 784004 p^{6} T^{6} - 556 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 512 T + 447916 T^{2} + 294997504 T^{3} + 154286433686 T^{4} + 294997504 p^{3} T^{5} + 447916 p^{6} T^{6} + 512 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 266 T + 796080 T^{2} - 331469650 T^{3} + 317353948286 T^{4} - 331469650 p^{3} T^{5} + 796080 p^{6} T^{6} - 266 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 320 T + 872396 T^{2} + 298640000 T^{3} + 431192413382 T^{4} + 298640000 p^{3} T^{5} + 872396 p^{6} T^{6} + 320 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 184 T + 1304636 T^{2} + 44316328 T^{3} + 763974499782 T^{4} + 44316328 p^{3} T^{5} + 1304636 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 728 T + 1800172 T^{2} + 877937112 T^{3} + 1366650646358 T^{4} + 877937112 p^{3} T^{5} + 1800172 p^{6} T^{6} + 728 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1714 T + 3032264 T^{2} + 3250729446 T^{3} + 3273084077934 T^{4} + 3250729446 p^{3} T^{5} + 3032264 p^{6} T^{6} + 1714 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 1216 T + 1986988 T^{2} + 2570430976 T^{3} + 2279523902118 T^{4} + 2570430976 p^{3} T^{5} + 1986988 p^{6} T^{6} + 1216 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.72344578033279303402207733606, −6.32554651718595644719335991460, −6.12509175002578024016620572587, −6.00249574798114703633328389369, −5.74196994877142653306547624411, −5.40352200876880191449689421224, −5.39142530643435378551823792926, −5.20856994709346462979192338616, −4.91325562936836703024593853676, −4.37570851370374354818308796735, −4.30097965598621896902469205606, −4.19217815252079043365691716308, −3.95281004613972614304364251100, −3.68400923997274970494523707610, −3.55802638531268842840807815501, −3.29294038812052834499562400860, −3.21640327338531343147373553254, −2.60375974823397939708462908599, −2.55080008258381601074418337345, −2.32264970773816447494235425677, −2.23689704995733681999829263422, −1.37341928564169608837669204589, −1.34989289555843907745368640470, −1.13165164653439912611461590690, −0.991373409008445146333456740339, 0, 0, 0, 0, 0.991373409008445146333456740339, 1.13165164653439912611461590690, 1.34989289555843907745368640470, 1.37341928564169608837669204589, 2.23689704995733681999829263422, 2.32264970773816447494235425677, 2.55080008258381601074418337345, 2.60375974823397939708462908599, 3.21640327338531343147373553254, 3.29294038812052834499562400860, 3.55802638531268842840807815501, 3.68400923997274970494523707610, 3.95281004613972614304364251100, 4.19217815252079043365691716308, 4.30097965598621896902469205606, 4.37570851370374354818308796735, 4.91325562936836703024593853676, 5.20856994709346462979192338616, 5.39142530643435378551823792926, 5.40352200876880191449689421224, 5.74196994877142653306547624411, 6.00249574798114703633328389369, 6.12509175002578024016620572587, 6.32554651718595644719335991460, 6.72344578033279303402207733606

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