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Properties

Label 8-1210e4-1.1-c3e4-0-3
Degree $8$
Conductor $2.144\times 10^{12}$
Sign $1$
Analytic cond. $2.59780\times 10^{7}$
Root an. cond. $8.44939$
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 − 6·3-s + 40·4-s + 20·5-s − 48·6-s − 25·7-s + 160·8-s − 50·9-s + 160·10-s − 240·12-s − 59·13-s − 200·14-s − 120·15-s + 560·16-s − 23·17-s − 400·18-s + 134·19-s + 800·20-s + 150·21-s − 259·23-s − 960·24-s + 250·25-s − 472·26-s + 373·27-s − 1.00e3·28-s − 236·29-s − 960·30-s + ⋯
L(s)  = 1  + 2.82·2-s − 1.15·3-s + 5·4-s + 1.78·5-s − 3.26·6-s − 1.34·7-s + 7.07·8-s − 1.85·9-s + 5.05·10-s − 5.77·12-s − 1.25·13-s − 3.81·14-s − 2.06·15-s + 35/4·16-s − 0.328·17-s − 5.23·18-s + 1.61·19-s + 8.94·20-s + 1.55·21-s − 2.34·23-s − 8.16·24-s + 2·25-s − 3.56·26-s + 2.65·27-s − 6.74·28-s − 1.51·29-s − 5.84·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.59780\times 10^{7}\)
Root analytic conductor: \(8.44939\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T + 86 T^{2} + 443 T^{3} + 3215 T^{4} + 443 p^{3} T^{5} + 86 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 211 p T^{2} + 25150 T^{3} + 778049 T^{4} + 25150 p^{3} T^{5} + 211 p^{7} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 59 T + 5359 T^{2} + 262688 T^{3} + 18437609 T^{4} + 262688 p^{3} T^{5} + 5359 p^{6} T^{6} + 59 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 14624 T^{2} + 336611 T^{3} + 98191175 T^{4} + 336611 p^{3} T^{5} + 14624 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 134 T + 20314 T^{2} - 1993871 T^{3} + 208165679 T^{4} - 1993871 p^{3} T^{5} + 20314 p^{6} T^{6} - 134 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 259 T + 58106 T^{2} + 9225487 T^{3} + 1114527805 T^{4} + 9225487 p^{3} T^{5} + 58106 p^{6} T^{6} + 259 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 236 T + 93260 T^{2} + 15215507 T^{3} + 118523227 p T^{4} + 15215507 p^{3} T^{5} + 93260 p^{6} T^{6} + 236 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 405 T + 158371 T^{2} + 37278030 T^{3} + 7664953741 T^{4} + 37278030 p^{3} T^{5} + 158371 p^{6} T^{6} + 405 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 55 T + 144960 T^{2} + 7172605 T^{3} + 10115014553 T^{4} + 7172605 p^{3} T^{5} + 144960 p^{6} T^{6} + 55 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 133 T + 52486 T^{2} + 26005053 T^{3} + 5125101239 T^{4} + 26005053 p^{3} T^{5} + 52486 p^{6} T^{6} + 133 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 147 T + 143539 T^{2} - 43083642 T^{3} + 11141772193 T^{4} - 43083642 p^{3} T^{5} + 143539 p^{6} T^{6} - 147 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 229 T + 145225 T^{2} + 12861584 T^{3} + 14185372251 T^{4} + 12861584 p^{3} T^{5} + 145225 p^{6} T^{6} + 229 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 303 T + 357809 T^{2} + 79847388 T^{3} + 70036744723 T^{4} + 79847388 p^{3} T^{5} + 357809 p^{6} T^{6} + 303 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 789 T + 696089 T^{2} + 335803986 T^{3} + 197262314179 T^{4} + 335803986 p^{3} T^{5} + 696089 p^{6} T^{6} + 789 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 468 T + 501695 T^{2} + 44932494 T^{3} + 85570037663 T^{4} + 44932494 p^{3} T^{5} + 501695 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 689 T + 890320 T^{2} + 304257839 T^{3} + 296128247201 T^{4} + 304257839 p^{3} T^{5} + 890320 p^{6} T^{6} + 689 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1499 T + 1791788 T^{2} + 1326542837 T^{3} + 924860219639 T^{4} + 1326542837 p^{3} T^{5} + 1791788 p^{6} T^{6} + 1499 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 1051 T + 1269146 T^{2} + 945254193 T^{3} + 732451583405 T^{4} + 945254193 p^{3} T^{5} + 1269146 p^{6} T^{6} + 1051 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1596 T + 2140020 T^{2} + 1899563187 T^{3} + 1569574851923 T^{4} + 1899563187 p^{3} T^{5} + 2140020 p^{6} T^{6} + 1596 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 1373 T + 2499507 T^{2} + 2366695020 T^{3} + 2204700059671 T^{4} + 2366695020 p^{3} T^{5} + 2499507 p^{6} T^{6} + 1373 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1684 T + 2956394 T^{2} + 3402078856 T^{3} + 3150321544379 T^{4} + 3402078856 p^{3} T^{5} + 2956394 p^{6} T^{6} + 1684 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1302 T + 1176644 T^{2} + 554950596 T^{3} - 842395114555 T^{4} + 554950596 p^{3} T^{5} + 1176644 p^{6} T^{6} - 1302 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.98103646688610669922122225649, −6.29090605739790959707391592842, −6.27341894685483511023529723730, −6.23617343626376652469928652930, −6.03642812994148138839019108035, −5.66958367192160563117447657434, −5.63157402321147244110944148480, −5.58360150775876041563482971655, −5.57316506187364765466200000342, −4.96300570805836059674359252980, −4.76451579749283701591750186336, −4.67177503273881627361176333978, −4.63989360098690560542071974298, −3.95003819952259048658934849751, −3.73303192700166607835239316065, −3.54077112428751125964790273479, −3.37163900501194250580115491162, −2.88289108834736714168408983234, −2.81067957627804754758221287024, −2.65403126152802991074339498866, −2.60080816400916928858334409270, −1.81063676726400278797992230959, −1.66869556639943906743943584794, −1.61073936490929327909545325067, −1.32269043112077199413625347380, 0, 0, 0, 0, 1.32269043112077199413625347380, 1.61073936490929327909545325067, 1.66869556639943906743943584794, 1.81063676726400278797992230959, 2.60080816400916928858334409270, 2.65403126152802991074339498866, 2.81067957627804754758221287024, 2.88289108834736714168408983234, 3.37163900501194250580115491162, 3.54077112428751125964790273479, 3.73303192700166607835239316065, 3.95003819952259048658934849751, 4.63989360098690560542071974298, 4.67177503273881627361176333978, 4.76451579749283701591750186336, 4.96300570805836059674359252980, 5.57316506187364765466200000342, 5.58360150775876041563482971655, 5.63157402321147244110944148480, 5.66958367192160563117447657434, 6.03642812994148138839019108035, 6.23617343626376652469928652930, 6.27341894685483511023529723730, 6.29090605739790959707391592842, 6.98103646688610669922122225649

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