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Properties

Label 8-8470e4-1.1-c1e4-0-0
Degree $8$
Conductor $5.147\times 10^{15}$
Sign $1$
Analytic cond. $2.09238\times 10^{7}$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 10·4-s + 4·5-s + 16·6-s − 4·7-s − 20·8-s + 5·9-s − 16·10-s − 40·12-s + 14·13-s + 16·14-s − 16·15-s + 35·16-s + 12·17-s − 20·18-s − 2·19-s + 40·20-s + 16·21-s + 80·24-s + 10·25-s − 56·26-s − 2·27-s − 40·28-s + 10·29-s + 64·30-s − 18·31-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 1.51·7-s − 7.07·8-s + 5/3·9-s − 5.05·10-s − 11.5·12-s + 3.88·13-s + 4.27·14-s − 4.13·15-s + 35/4·16-s + 2.91·17-s − 4.71·18-s − 0.458·19-s + 8.94·20-s + 3.49·21-s + 16.3·24-s + 2·25-s − 10.9·26-s − 0.384·27-s − 7.55·28-s + 1.85·29-s + 11.6·30-s − 3.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.09238\times 10^{7}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.909479474\)
\(L(\frac12)\) \(\approx\) \(1.909479474\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 11 T^{2} + 26 T^{3} + 53 T^{4} + 26 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 106 T^{2} - 42 p T^{3} + 2198 T^{4} - 42 p^{2} T^{5} + 106 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 109 T^{2} - 642 T^{3} + 3103 T^{4} - 642 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 47 T^{2} + 104 T^{3} + 1209 T^{4} + 104 p T^{5} + 47 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 60 T^{2} + 1878 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 78 T^{2} - 430 T^{3} + 2678 T^{4} - 430 p T^{5} + 78 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 138 T^{2} + 546 T^{3} + 1950 T^{4} + 546 p T^{5} + 138 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 194 T^{2} - 1558 T^{3} + 10518 T^{4} - 1558 p T^{5} + 194 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 367 T^{2} - 3660 T^{3} + 27571 T^{4} - 3660 p T^{5} + 367 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 119 T^{2} - 1010 T^{3} + 7507 T^{4} - 1010 p T^{5} + 119 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 120 T^{2} + 392 T^{3} + 5246 T^{4} + 392 p T^{5} + 120 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 290 T^{2} - 2680 T^{3} + 22318 T^{4} - 2680 p T^{5} + 290 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 215 T^{2} + 2188 T^{3} + 18253 T^{4} + 2188 p T^{5} + 215 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 260 T^{2} - 2406 T^{3} + 23994 T^{4} - 2406 p T^{5} + 260 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 245 T^{2} - 50 T^{3} + 23823 T^{4} - 50 p T^{5} + 245 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 322 T^{2} + 2830 T^{3} + 35086 T^{4} + 2830 p T^{5} + 322 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 395 T^{2} - 2990 T^{3} + 20763 T^{4} - 2990 p T^{5} + 395 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 242 T^{2} - 1596 T^{3} + 27174 T^{4} - 1596 p T^{5} + 242 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 203 T^{2} - 1142 T^{3} + 18865 T^{4} - 1142 p T^{5} + 203 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 187 T^{2} - 328 T^{3} + 14695 T^{4} - 328 p T^{5} + 187 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 311 T^{2} + 2150 T^{3} + 40087 T^{4} + 2150 p T^{5} + 311 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.65543757028184912859262079006, −5.55460900138522389593334353634, −5.39580343270807929005473190096, −5.20488413981916567854879708320, −5.18853095554908693437829449492, −4.45977790513896467129545797671, −4.20603597283053100972634224128, −4.15344644653867225805077516777, −3.91745160870878178667563781698, −3.77225918159377119369412292607, −3.41564152627358228739264780078, −3.27862373254277757481007035569, −3.15493343883613805990547867730, −2.69533593598287245645977617227, −2.65642367889862589615225037005, −2.53482307271622216214480836501, −2.23221532799221848469404404589, −1.75551143019542997734183868680, −1.66719804482218482819879359091, −1.44722134034854773190647822756, −1.08732303067700030086539952349, −0.878978692381912878813734382724, −0.794641884309409337643998605268, −0.60166164552959015683811491141, −0.47133589874212518218099991935, 0.47133589874212518218099991935, 0.60166164552959015683811491141, 0.794641884309409337643998605268, 0.878978692381912878813734382724, 1.08732303067700030086539952349, 1.44722134034854773190647822756, 1.66719804482218482819879359091, 1.75551143019542997734183868680, 2.23221532799221848469404404589, 2.53482307271622216214480836501, 2.65642367889862589615225037005, 2.69533593598287245645977617227, 3.15493343883613805990547867730, 3.27862373254277757481007035569, 3.41564152627358228739264780078, 3.77225918159377119369412292607, 3.91745160870878178667563781698, 4.15344644653867225805077516777, 4.20603597283053100972634224128, 4.45977790513896467129545797671, 5.18853095554908693437829449492, 5.20488413981916567854879708320, 5.39580343270807929005473190096, 5.55460900138522389593334353634, 5.65543757028184912859262079006

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