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Properties

Label 8-672e4-1.1-c3e4-0-1
Degree $8$
Conductor $203928109056$
Sign $1$
Analytic cond. $2.47139\times 10^{6}$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 12·5-s − 36·7-s + 9·9-s + 46·11-s + 88·13-s − 72·15-s − 12·17-s + 84·19-s + 216·21-s + 148·23-s + 249·25-s + 54·27-s − 152·29-s + 140·31-s − 276·33-s − 432·35-s + 700·37-s − 528·39-s − 608·41-s + 696·43-s + 108·45-s − 784·47-s + 323·49-s + 72·51-s − 244·53-s + 552·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.07·5-s − 1.94·7-s + 1/3·9-s + 1.26·11-s + 1.87·13-s − 1.23·15-s − 0.171·17-s + 1.01·19-s + 2.24·21-s + 1.34·23-s + 1.99·25-s + 0.384·27-s − 0.973·29-s + 0.811·31-s − 1.45·33-s − 2.08·35-s + 3.11·37-s − 2.16·39-s − 2.31·41-s + 2.46·43-s + 0.357·45-s − 2.43·47-s + 0.941·49-s + 0.197·51-s − 0.632·53-s + 1.35·55-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2.47139\times 10^{6}\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.181380134\)
\(L(\frac12)\) \(\approx\) \(3.181380134\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 36 T + 139 p T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
good5$D_4\times C_2$ \( 1 - 12 T - 21 p T^{2} + 12 T^{3} + 28376 T^{4} + 12 p^{3} T^{5} - 21 p^{7} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 46 T - 483 T^{2} + 2898 T^{3} + 2166844 T^{4} + 2898 p^{3} T^{5} - 483 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 44 T + 4730 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 12 T - 354 p T^{2} - 43968 T^{3} + 13125203 T^{4} - 43968 p^{3} T^{5} - 354 p^{7} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 84 T - 8278 T^{2} - 135744 T^{3} + 139688571 T^{4} - 135744 p^{3} T^{5} - 8278 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 148 T + 10002 T^{2} + 1839936 T^{3} - 272884253 T^{4} + 1839936 p^{3} T^{5} + 10002 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 76 T + 27097 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 140 T - 43069 T^{2} - 432180 T^{3} + 2455996280 T^{4} - 432180 p^{3} T^{5} - 43069 p^{6} T^{6} - 140 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 700 T + 269894 T^{2} - 83160000 T^{3} + 21060954827 T^{4} - 83160000 p^{3} T^{5} + 269894 p^{6} T^{6} - 700 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 304 T + 101746 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 348 T + 188698 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 784 T + 306774 T^{2} + 78585024 T^{3} + 20196680707 T^{4} + 78585024 p^{3} T^{5} + 306774 p^{6} T^{6} + 784 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 244 T - 251289 T^{2} + 3189324 T^{3} + 65584278424 T^{4} + 3189324 p^{3} T^{5} - 251289 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 54 T - 147499 T^{2} + 14058522 T^{3} - 20154746580 T^{4} + 14058522 p^{3} T^{5} - 147499 p^{6} T^{6} - 54 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 416 T - 216278 T^{2} - 26885248 T^{3} + 60780165259 T^{4} - 26885248 p^{3} T^{5} - 216278 p^{6} T^{6} + 416 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 1176 T + 435854 T^{2} + 406420896 T^{3} + 390633690027 T^{4} + 406420896 p^{3} T^{5} + 435854 p^{6} T^{6} + 1176 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 392 T + 8870 p T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 1676 T + 1400330 T^{2} + 1056905712 T^{3} + 740209910435 T^{4} + 1056905712 p^{3} T^{5} + 1400330 p^{6} T^{6} + 1676 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 20 T - 754861 T^{2} - 4616340 T^{3} + 327134507000 T^{4} - 4616340 p^{3} T^{5} - 754861 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 78 T + 953287 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 16 T - 390174 T^{2} + 16312128 T^{3} - 344726178125 T^{4} + 16312128 p^{3} T^{5} - 390174 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 86 T + 405803 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−7.02010377786748199773279501206, −6.68402857734123351862954490483, −6.49715484678588691117913209884, −6.46212483294796060137162797858, −6.11409967418819861505093232381, −5.94277596150990159887809064169, −5.92096853533053274932793086140, −5.77847131629975191738075571916, −5.20000853896704376843254226042, −5.03626732794383630682897267445, −4.66290815565460831618213432737, −4.59404015461322004225014414304, −4.24119082117047635874320667816, −3.84394365629814917799797568344, −3.54815940152912408363728506261, −3.24757015369762865330398998699, −3.04298726771870359825641169088, −2.90074323405699673562337769293, −2.63336610710500226958590268639, −1.92458170480073056176146774737, −1.50424341220620530426373223590, −1.42359392455585130053457967793, −0.840791838420509280135393503663, −0.810869348430025628667829548286, −0.28918888583030975678977169893, 0.28918888583030975678977169893, 0.810869348430025628667829548286, 0.840791838420509280135393503663, 1.42359392455585130053457967793, 1.50424341220620530426373223590, 1.92458170480073056176146774737, 2.63336610710500226958590268639, 2.90074323405699673562337769293, 3.04298726771870359825641169088, 3.24757015369762865330398998699, 3.54815940152912408363728506261, 3.84394365629814917799797568344, 4.24119082117047635874320667816, 4.59404015461322004225014414304, 4.66290815565460831618213432737, 5.03626732794383630682897267445, 5.20000853896704376843254226042, 5.77847131629975191738075571916, 5.92096853533053274932793086140, 5.94277596150990159887809064169, 6.11409967418819861505093232381, 6.46212483294796060137162797858, 6.49715484678588691117913209884, 6.68402857734123351862954490483, 7.02010377786748199773279501206

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