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Properties

Label 8-3750e4-1.1-c1e4-0-6
Degree $8$
Conductor $1.978\times 10^{14}$
Sign $1$
Analytic cond. $803958.$
Root an. cond. $5.47210$
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 − 16·6-s + 4·7-s − 20·8-s + 10·9-s + 40·12-s − 2·13-s − 16·14-s + 35·16-s + 9·17-s − 40·18-s − 11·19-s + 16·21-s + 15·23-s − 80·24-s + 8·26-s + 20·27-s + 40·28-s + 6·29-s − 8·31-s − 56·32-s − 36·34-s + 100·36-s + 13·37-s + 44·38-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s − 6.53·6-s + 1.51·7-s − 7.07·8-s + 10/3·9-s + 11.5·12-s − 0.554·13-s − 4.27·14-s + 35/4·16-s + 2.18·17-s − 9.42·18-s − 2.52·19-s + 3.49·21-s + 3.12·23-s − 16.3·24-s + 1.56·26-s + 3.84·27-s + 7.55·28-s + 1.11·29-s − 1.43·31-s − 9.89·32-s − 6.17·34-s + 50/3·36-s + 2.13·37-s + 7.13·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(803958.\)
Root analytic conductor: \(5.47210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.726628057\)
\(L(\frac12)\) \(\approx\) \(9.726628057\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
good7$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 24 T^{2} - 83 T^{3} + 239 T^{4} - 83 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 + 29 T^{2} + 441 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 36 T^{2} + 76 T^{3} + 629 T^{4} + 76 p T^{5} + 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 89 T^{2} - 468 T^{3} + 2439 T^{4} - 468 p T^{5} + 89 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 11 T + 102 T^{2} + 613 T^{3} + 3155 T^{4} + 613 p T^{5} + 102 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 152 T^{2} - 45 p T^{3} + 5709 T^{4} - 45 p^{2} T^{5} + 152 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 92 T^{2} - 468 T^{3} + 3645 T^{4} - 468 p T^{5} + 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 123 T^{2} + 706 T^{3} + 5675 T^{4} + 706 p T^{5} + 123 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 177 T^{2} - 1460 T^{3} + 10361 T^{4} - 1460 p T^{5} + 177 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 35 T^{2} + 396 T^{3} - 2391 T^{4} + 396 p T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 108 T^{2} - 515 T^{3} + 5801 T^{4} - 515 p T^{5} + 108 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 158 T^{2} - 1125 T^{3} + 7779 T^{4} - 1125 p T^{5} + 158 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 21 T + 338 T^{2} - 3555 T^{3} + 30291 T^{4} - 3555 p T^{5} + 338 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 221 T^{2} - 1800 T^{3} + 279 p T^{4} - 1800 p T^{5} + 221 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 174 T^{2} + 1105 T^{3} + 13631 T^{4} + 1105 p T^{5} + 174 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 19 T + 309 T^{2} - 3578 T^{3} + 32609 T^{4} - 3578 p T^{5} + 309 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 224 T^{2} + 45 T^{3} + 21771 T^{4} + 45 p T^{5} + 224 p^{2} T^{6} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 96 T^{2} + 991 T^{3} + 4529 T^{4} + 991 p T^{5} + 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 390 T^{2} + 4183 T^{3} + 42269 T^{4} + 4183 p T^{5} + 390 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 18 T + 416 T^{2} - 4509 T^{3} + 54999 T^{4} - 4509 p T^{5} + 416 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 27 T + 620 T^{2} - 8307 T^{3} + 96129 T^{4} - 8307 p T^{5} + 620 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 192 T^{2} - 875 T^{3} + 9611 T^{4} - 875 p T^{5} + 192 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.37213987145661110268895534345, −5.71599934927819791055069077852, −5.69513713042707830924809613629, −5.61332673687601167498700471768, −5.35302428542919072390453181336, −4.75482266198918595574705374838, −4.72213401282573685579241742510, −4.64434056493548161889214237363, −4.62439067847869370814860064637, −3.90844047692170492955816511168, −3.76544480608072051631207075364, −3.61553395258646265915848371339, −3.60455532756006951030381792546, −3.03007129230299101109942587339, −2.88556905671616012570440214887, −2.64102238922756723710437179395, −2.60877801160057254051333367738, −2.18887207981703329902781177274, −1.97866004340215080045084800956, −1.96481171434495966406501134647, −1.73533116010023510428199667088, −1.01140215242727644815397766761, −0.979128262815176401015477450972, −0.796807578777940509871175178271, −0.69237348451627590045687321964, 0.69237348451627590045687321964, 0.796807578777940509871175178271, 0.979128262815176401015477450972, 1.01140215242727644815397766761, 1.73533116010023510428199667088, 1.96481171434495966406501134647, 1.97866004340215080045084800956, 2.18887207981703329902781177274, 2.60877801160057254051333367738, 2.64102238922756723710437179395, 2.88556905671616012570440214887, 3.03007129230299101109942587339, 3.60455532756006951030381792546, 3.61553395258646265915848371339, 3.76544480608072051631207075364, 3.90844047692170492955816511168, 4.62439067847869370814860064637, 4.64434056493548161889214237363, 4.72213401282573685579241742510, 4.75482266198918595574705374838, 5.35302428542919072390453181336, 5.61332673687601167498700471768, 5.69513713042707830924809613629, 5.71599934927819791055069077852, 6.37213987145661110268895534345

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