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Properties

Label 8-1890e4-1.1-c3e4-0-4
Degree $8$
Conductor $1.276\times 10^{13}$
Sign $1$
Analytic cond. $1.54636\times 10^{8}$
Root an. cond. $10.5600$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 40·4-s + 20·5-s + 28·7-s + 160·8-s + 160·10-s + 37·11-s + 54·13-s + 224·14-s + 560·16-s + 110·17-s + 125·19-s + 800·20-s + 296·22-s + 4·23-s + 250·25-s + 432·26-s + 1.12e3·28-s + 195·29-s + 60·31-s + 1.79e3·32-s + 880·34-s + 560·35-s + 123·37-s + 1.00e3·38-s + 3.20e3·40-s − 16·41-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 1.78·5-s + 1.51·7-s + 7.07·8-s + 5.05·10-s + 1.01·11-s + 1.15·13-s + 4.27·14-s + 35/4·16-s + 1.56·17-s + 1.50·19-s + 8.94·20-s + 2.86·22-s + 0.0362·23-s + 2·25-s + 3.25·26-s + 7.55·28-s + 1.24·29-s + 0.347·31-s + 9.89·32-s + 4.43·34-s + 2.70·35-s + 0.546·37-s + 4.26·38-s + 12.6·40-s − 0.0609·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.54636\times 10^{8}\)
Root analytic conductor: \(10.5600\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(455.1575730\)
\(L(\frac12)\) \(\approx\) \(455.1575730\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{4} \)
7$C_1$ \( ( 1 - p T )^{4} \)
good11$C_2 \wr S_4$ \( 1 - 37 T + 2517 T^{2} - 80976 T^{3} + 4137610 T^{4} - 80976 p^{3} T^{5} + 2517 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 54 T + 5300 T^{2} - 116901 T^{3} + 10891962 T^{4} - 116901 p^{3} T^{5} + 5300 p^{6} T^{6} - 54 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 110 T + 12768 T^{2} - 1074801 T^{3} + 101229052 T^{4} - 1074801 p^{3} T^{5} + 12768 p^{6} T^{6} - 110 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 125 T + 20233 T^{2} - 1835525 T^{3} + 205640728 T^{4} - 1835525 p^{3} T^{5} + 20233 p^{6} T^{6} - 125 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 4 T + 12570 T^{2} - 1492119 T^{3} + 114229480 T^{4} - 1492119 p^{3} T^{5} + 12570 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 195 T + 62330 T^{2} - 11985390 T^{3} + 2129329743 T^{4} - 11985390 p^{3} T^{5} + 62330 p^{6} T^{6} - 195 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 60 T + 29534 T^{2} + 2847120 T^{3} + 712903551 T^{4} + 2847120 p^{3} T^{5} + 29534 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 123 T + 102533 T^{2} - 15484314 T^{3} + 5619691878 T^{4} - 15484314 p^{3} T^{5} + 102533 p^{6} T^{6} - 123 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 16 T + 1938 p T^{2} + 29617557 T^{3} + 3031049794 T^{4} + 29617557 p^{3} T^{5} + 1938 p^{7} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 211 T - 51469 T^{2} + 15143904 T^{3} - 1174475104 T^{4} + 15143904 p^{3} T^{5} - 51469 p^{6} T^{6} - 211 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 268 T + 277494 T^{2} - 66236496 T^{3} + 39765686935 T^{4} - 66236496 p^{3} T^{5} + 277494 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 9 T + 183011 T^{2} + 47458953 T^{3} + 12851053800 T^{4} + 47458953 p^{3} T^{5} + 183011 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 643 T + 636843 T^{2} - 280531524 T^{3} + 174144057148 T^{4} - 280531524 p^{3} T^{5} + 636843 p^{6} T^{6} - 643 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 318 T + 799112 T^{2} - 208853979 T^{3} + 260304495150 T^{4} - 208853979 p^{3} T^{5} + 799112 p^{6} T^{6} - 318 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 289 T + 777377 T^{2} - 57880515 T^{3} + 264497093948 T^{4} - 57880515 p^{3} T^{5} + 777377 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 236 T + 1360938 T^{2} - 244139607 T^{3} + 718629616594 T^{4} - 244139607 p^{3} T^{5} + 1360938 p^{6} T^{6} - 236 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 627 T + 1375775 T^{2} - 692034474 T^{3} + 775727834148 T^{4} - 692034474 p^{3} T^{5} + 1375775 p^{6} T^{6} - 627 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 130 T + 1709642 T^{2} - 129337245 T^{3} + 1195946452154 T^{4} - 129337245 p^{3} T^{5} + 1709642 p^{6} T^{6} - 130 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 120 T + 1572143 T^{2} + 377930520 T^{3} + 1158940192344 T^{4} + 377930520 p^{3} T^{5} + 1572143 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 644 T + 1489569 T^{2} - 1231320078 T^{3} + 1349683303000 T^{4} - 1231320078 p^{3} T^{5} + 1489569 p^{6} T^{6} - 644 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1053 T + 1622435 T^{2} - 1811718333 T^{3} + 2472528930612 T^{4} - 1811718333 p^{3} T^{5} + 1622435 p^{6} T^{6} - 1053 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.17173949962242356633495476723, −5.68727301057237717633015125273, −5.60950337802003641792409426396, −5.60923614242590633631632463899, −5.49366981601887100479283202800, −4.93619553123735089885375707927, −4.88298334666453623353942853675, −4.84149726517235594263791453938, −4.72502497692933157309722067376, −4.03976491903125632311538646119, −3.97074186081479043644571004423, −3.96198520344671276710275229012, −3.70469379550349015639596315651, −3.28804270903613561738084180416, −2.95270314356376284414137839707, −2.90713971215590283487628710018, −2.85716345170907325931956902941, −2.11953636618769809035288385280, −2.07400167201349585750302861153, −1.97221810385302203472946455810, −1.69746705861149172010435999255, −1.15077081068524437924005191927, −0.953946829351813042238215005113, −0.874284354214068524001016413293, −0.853111201479732684852158012479, 0.853111201479732684852158012479, 0.874284354214068524001016413293, 0.953946829351813042238215005113, 1.15077081068524437924005191927, 1.69746705861149172010435999255, 1.97221810385302203472946455810, 2.07400167201349585750302861153, 2.11953636618769809035288385280, 2.85716345170907325931956902941, 2.90713971215590283487628710018, 2.95270314356376284414137839707, 3.28804270903613561738084180416, 3.70469379550349015639596315651, 3.96198520344671276710275229012, 3.97074186081479043644571004423, 4.03976491903125632311538646119, 4.72502497692933157309722067376, 4.84149726517235594263791453938, 4.88298334666453623353942853675, 4.93619553123735089885375707927, 5.49366981601887100479283202800, 5.60923614242590633631632463899, 5.60950337802003641792409426396, 5.68727301057237717633015125273, 6.17173949962242356633495476723

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