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Properties

Label 8-648e4-1.1-c3e4-0-9
Degree $8$
Conductor $176319369216$
Sign $1$
Analytic cond. $2.13680\times 10^{6}$
Root an. cond. $6.18330$
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  − 5·5-s − 3·7-s + 16·11-s − 29·13-s + 17·17-s − 109·19-s + 37·23-s − 286·25-s − 3·29-s − 331·31-s + 15·35-s − 366·37-s − 378·41-s − 506·43-s + 171·47-s − 1.09e3·49-s − 410·53-s − 80·55-s + 616·59-s − 1.33e3·61-s + 145·65-s − 1.16e3·67-s + 344·71-s − 1.30e3·73-s − 48·77-s − 1.85e3·79-s + 1.42e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.161·7-s + 0.438·11-s − 0.618·13-s + 0.242·17-s − 1.31·19-s + 0.335·23-s − 2.28·25-s − 0.0192·29-s − 1.91·31-s + 0.0724·35-s − 1.62·37-s − 1.43·41-s − 1.79·43-s + 0.530·47-s − 3.19·49-s − 1.06·53-s − 0.196·55-s + 1.35·59-s − 2.79·61-s + 0.276·65-s − 2.11·67-s + 0.575·71-s − 2.09·73-s − 0.0710·77-s − 2.63·79-s + 1.87·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.13680\times 10^{6}\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{16} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
good5$C_2 \wr S_4$ \( 1 + p T + 311 T^{2} + 2378 T^{3} + 47908 T^{4} + 2378 p^{3} T^{5} + 311 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 3 T + 1105 T^{2} + 2700 T^{3} + 523326 T^{4} + 2700 p^{3} T^{5} + 1105 p^{6} T^{6} + 3 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 16 T + 3458 T^{2} - 65944 T^{3} + 518273 p T^{4} - 65944 p^{3} T^{5} + 3458 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 29 T + 5017 T^{2} + 224474 T^{3} + 12532342 T^{4} + 224474 p^{3} T^{5} + 5017 p^{6} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - p T + 5090 T^{2} + 472705 T^{3} + 4306426 T^{4} + 472705 p^{3} T^{5} + 5090 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 109 T + 15664 T^{2} + 996349 T^{3} + 115705582 T^{4} + 996349 p^{3} T^{5} + 15664 p^{6} T^{6} + 109 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 37 T + 29813 T^{2} - 1074868 T^{3} + 467349442 T^{4} - 1074868 p^{3} T^{5} + 29813 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 3 T + 3149 T^{2} + 3145950 T^{3} + 536696802 T^{4} + 3145950 p^{3} T^{5} + 3149 p^{6} T^{6} + 3 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 331 T + 52015 T^{2} - 5036396 T^{3} - 1858547228 T^{4} - 5036396 p^{3} T^{5} + 52015 p^{6} T^{6} + 331 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 366 T + 184120 T^{2} + 47971818 T^{3} + 13305026046 T^{4} + 47971818 p^{3} T^{5} + 184120 p^{6} T^{6} + 366 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 378 T + 139688 T^{2} + 40094532 T^{3} + 9393845481 T^{4} + 40094532 p^{3} T^{5} + 139688 p^{6} T^{6} + 378 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 506 T + 249700 T^{2} + 75502472 T^{3} + 26495780485 T^{4} + 75502472 p^{3} T^{5} + 249700 p^{6} T^{6} + 506 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 171 T + 378233 T^{2} - 45800820 T^{3} + 56669444838 T^{4} - 45800820 p^{3} T^{5} + 378233 p^{6} T^{6} - 171 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 410 T + 355928 T^{2} + 45836174 T^{3} + 45687169918 T^{4} + 45836174 p^{3} T^{5} + 355928 p^{6} T^{6} + 410 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 616 T + 335522 T^{2} - 2358200 p T^{3} + 54872410027 T^{4} - 2358200 p^{4} T^{5} + 335522 p^{6} T^{6} - 616 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1331 T + 1257583 T^{2} + 772644974 T^{3} + 6880956220 p T^{4} + 772644974 p^{3} T^{5} + 1257583 p^{6} T^{6} + 1331 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1162 T + 873112 T^{2} + 389858584 T^{3} + 206962604953 T^{4} + 389858584 p^{3} T^{5} + 873112 p^{6} T^{6} + 1162 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 344 T + 777260 T^{2} - 172546424 T^{3} + 309941629126 T^{4} - 172546424 p^{3} T^{5} + 777260 p^{6} T^{6} - 344 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1307 T + 1338958 T^{2} + 911507069 T^{3} + 651081152866 T^{4} + 911507069 p^{3} T^{5} + 1338958 p^{6} T^{6} + 1307 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1853 T + 2847415 T^{2} + 2649679196 T^{3} + 2222927534284 T^{4} + 2649679196 p^{3} T^{5} + 2847415 p^{6} T^{6} + 1853 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1421 T + 2006375 T^{2} - 1942496552 T^{3} + 1708446285340 T^{4} - 1942496552 p^{3} T^{5} + 2006375 p^{6} T^{6} - 1421 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 816 T + 2512220 T^{2} + 1638983952 T^{3} + 2570115837030 T^{4} + 1638983952 p^{3} T^{5} + 2512220 p^{6} T^{6} + 816 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2506 T + 4515844 T^{2} + 5331969220 T^{3} + 5758383832309 T^{4} + 5331969220 p^{3} T^{5} + 4515844 p^{6} T^{6} + 2506 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.66877804297434118765627876147, −7.46124561627944787166588387352, −7.10080978797935277254534299192, −7.01834854634284440631215690744, −6.73454345798444295192962265484, −6.36259795309011075337852329160, −6.33660252804266417429609327988, −6.08644449127345988066175931336, −5.80096249330672866898354371667, −5.28197025791286156375990825500, −5.25939492838950541430284822445, −5.05309093284091555544482921537, −4.88800211756213912341560766664, −4.21389087933918221652191692428, −4.19945171521150039823917148651, −3.92357669614289107553964477703, −3.81359204338220403816588593512, −3.27872672187186674966968593628, −3.14222434354455286557490419559, −2.70146159098039827452231848844, −2.67308267044506813294836534845, −1.76687586863298064772537913840, −1.75109175635527363113185441429, −1.51904171903100773035616475147, −1.39313532058753252090211712218, 0, 0, 0, 0, 1.39313532058753252090211712218, 1.51904171903100773035616475147, 1.75109175635527363113185441429, 1.76687586863298064772537913840, 2.67308267044506813294836534845, 2.70146159098039827452231848844, 3.14222434354455286557490419559, 3.27872672187186674966968593628, 3.81359204338220403816588593512, 3.92357669614289107553964477703, 4.19945171521150039823917148651, 4.21389087933918221652191692428, 4.88800211756213912341560766664, 5.05309093284091555544482921537, 5.25939492838950541430284822445, 5.28197025791286156375990825500, 5.80096249330672866898354371667, 6.08644449127345988066175931336, 6.33660252804266417429609327988, 6.36259795309011075337852329160, 6.73454345798444295192962265484, 7.01834854634284440631215690744, 7.10080978797935277254534299192, 7.46124561627944787166588387352, 7.66877804297434118765627876147

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