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Properties

Label 8-2178e4-1.1-c3e4-0-6
Degree $8$
Conductor $2.250\times 10^{13}$
Sign $1$
Analytic cond. $2.72706\times 10^{8}$
Root an. cond. $11.3360$
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  − 8·2-s + 40·4-s − 20·5-s + 21·7-s − 160·8-s + 160·10-s + 38·13-s − 168·14-s + 560·16-s − 85·17-s + 5·19-s − 800·20-s − 316·23-s + 109·25-s − 304·26-s + 840·28-s − 123·29-s + 436·31-s − 1.79e3·32-s + 680·34-s − 420·35-s + 336·37-s − 40·38-s + 3.20e3·40-s + 394·41-s − 680·43-s + 2.52e3·46-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 1.78·5-s + 1.13·7-s − 7.07·8-s + 5.05·10-s + 0.810·13-s − 3.20·14-s + 35/4·16-s − 1.21·17-s + 0.0603·19-s − 8.94·20-s − 2.86·23-s + 0.871·25-s − 2.29·26-s + 5.66·28-s − 0.787·29-s + 2.52·31-s − 9.89·32-s + 3.42·34-s − 2.02·35-s + 1.49·37-s − 0.170·38-s + 12.6·40-s + 1.50·41-s − 2.41·43-s + 8.10·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.72706\times 10^{8}\)
Root analytic conductor: \(11.3360\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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$C_1$ \( ( 1 + p T )^{4} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 p T + 291 T^{2} + 858 p T^{3} + 63159 T^{4} + 858 p^{4} T^{5} + 291 p^{6} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 p T + 1062 T^{2} - 17552 T^{3} + 513663 T^{4} - 17552 p^{3} T^{5} + 1062 p^{6} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 38 T + 5631 T^{2} - 198736 T^{3} + 16696232 T^{4} - 198736 p^{3} T^{5} + 5631 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 5 p T + 17941 T^{2} + 1133735 T^{3} + 130328472 T^{4} + 1133735 p^{3} T^{5} + 17941 p^{6} T^{6} + 5 p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 20951 T^{2} - 328935 T^{3} + 191632156 T^{4} - 328935 p^{3} T^{5} + 20951 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 316 T + 70009 T^{2} + 10934882 T^{3} + 1406215104 T^{4} + 10934882 p^{3} T^{5} + 70009 p^{6} T^{6} + 316 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 123 T + 35221 T^{2} - 1869123 T^{3} + 117220020 T^{4} - 1869123 p^{3} T^{5} + 35221 p^{6} T^{6} + 123 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 436 T + 135561 T^{2} - 30858892 T^{3} + 6021643001 T^{4} - 30858892 p^{3} T^{5} + 135561 p^{6} T^{6} - 436 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 336 T + 126287 T^{2} - 20085192 T^{3} + 6173804228 T^{4} - 20085192 p^{3} T^{5} + 126287 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 394 T + 283151 T^{2} - 72337768 T^{3} + 29036356376 T^{4} - 72337768 p^{3} T^{5} + 283151 p^{6} T^{6} - 394 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 680 T + 443037 T^{2} + 165342210 T^{3} + 57688964824 T^{4} + 165342210 p^{3} T^{5} + 443037 p^{6} T^{6} + 680 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 1085 T + 616651 T^{2} + 232048825 T^{3} + 75928011672 T^{4} + 232048825 p^{3} T^{5} + 616651 p^{6} T^{6} + 1085 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 880 T + 259213 T^{2} - 97128920 T^{3} - 80706941631 T^{4} - 97128920 p^{3} T^{5} + 259213 p^{6} T^{6} + 880 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 724 T + 121861 T^{2} - 49833860 T^{3} - 22443603015 T^{4} - 49833860 p^{3} T^{5} + 121861 p^{6} T^{6} + 724 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 355 T + 206563 T^{2} - 100030015 T^{3} + 111325909928 T^{4} - 100030015 p^{3} T^{5} + 206563 p^{6} T^{6} - 355 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 869 T + 1113149 T^{2} + 573590277 T^{3} + 449410682092 T^{4} + 573590277 p^{3} T^{5} + 1113149 p^{6} T^{6} + 869 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 331 T + 752855 T^{2} - 484489279 T^{3} + 277177921244 T^{4} - 484489279 p^{3} T^{5} + 752855 p^{6} T^{6} - 331 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 1135 T + 1926189 T^{2} - 1347414995 T^{3} + 1193673823172 T^{4} - 1347414995 p^{3} T^{5} + 1926189 p^{6} T^{6} - 1135 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 1083 T + 881864 T^{2} - 166253544 T^{3} + 120819513641 T^{4} - 166253544 p^{3} T^{5} + 881864 p^{6} T^{6} - 1083 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 169 T + 1847318 T^{2} - 334846216 T^{3} + 1459517237603 T^{4} - 334846216 p^{3} T^{5} + 1847318 p^{6} T^{6} - 169 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 258 T + 203191 T^{2} - 25747352 T^{3} + 476125737520 T^{4} - 25747352 p^{3} T^{5} + 203191 p^{6} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 248 T + 3208071 T^{2} + 578108334 T^{3} + 4184809173679 T^{4} + 578108334 p^{3} T^{5} + 3208071 p^{6} T^{6} + 248 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.61092404218735661385607485858, −6.27773079297413629745602591974, −6.17871470320578740802468387671, −6.15597952401815853154636196106, −6.09783034573007671253879577852, −5.47910173373825380630114099670, −5.12553417237832862525395729672, −5.10919473638988290702538889623, −4.77421189054532036516708451368, −4.41942001667981019618302051862, −4.40297850189619770213841651274, −4.10885928443221397348856949510, −4.01134004924249387562552868787, −3.37732287720934389682164790073, −3.26394745330261674578801684387, −3.21177241216410540259852317429, −3.13832287133698409318842007126, −2.37744775795134452039045159097, −2.26986125137035000794345843616, −2.02781449349559451195388892134, −1.88988053026649869168680314098, −1.48485363045770181211517285788, −1.34916748876745691210044053429, −0.952570611711983742898833563204, −0.813134516965456818050532924069, 0, 0, 0, 0, 0.813134516965456818050532924069, 0.952570611711983742898833563204, 1.34916748876745691210044053429, 1.48485363045770181211517285788, 1.88988053026649869168680314098, 2.02781449349559451195388892134, 2.26986125137035000794345843616, 2.37744775795134452039045159097, 3.13832287133698409318842007126, 3.21177241216410540259852317429, 3.26394745330261674578801684387, 3.37732287720934389682164790073, 4.01134004924249387562552868787, 4.10885928443221397348856949510, 4.40297850189619770213841651274, 4.41942001667981019618302051862, 4.77421189054532036516708451368, 5.10919473638988290702538889623, 5.12553417237832862525395729672, 5.47910173373825380630114099670, 6.09783034573007671253879577852, 6.15597952401815853154636196106, 6.17871470320578740802468387671, 6.27773079297413629745602591974, 6.61092404218735661385607485858

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