Properties

Label 8-980e4-1.1-c3e4-0-7
Degree $8$
Conductor $922368160000$
Sign $1$
Analytic cond. $1.11781\times 10^{7}$
Root an. cond. $7.60406$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 10·5-s + 17·9-s − 20·11-s − 216·13-s − 60·15-s + 180·17-s − 64·19-s − 22·23-s + 25·25-s + 234·27-s + 956·29-s − 540·31-s + 120·33-s + 176·37-s + 1.29e3·39-s − 92·41-s − 508·43-s + 170·45-s + 508·47-s − 1.08e3·51-s + 344·53-s − 200·55-s + 384·57-s + 184·59-s − 94·61-s − 2.16e3·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.629·9-s − 0.548·11-s − 4.60·13-s − 1.03·15-s + 2.56·17-s − 0.772·19-s − 0.199·23-s + 1/5·25-s + 1.66·27-s + 6.12·29-s − 3.12·31-s + 0.633·33-s + 0.782·37-s + 5.32·39-s − 0.350·41-s − 1.80·43-s + 0.563·45-s + 1.57·47-s − 2.96·51-s + 0.891·53-s − 0.490·55-s + 0.892·57-s + 0.406·59-s − 0.197·61-s − 4.12·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.11781\times 10^{7}\)
Root analytic conductor: \(7.60406\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.472458085\)
\(L(\frac12)\) \(\approx\) \(4.472458085\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 2 p T + 19 T^{2} - 74 p T^{3} - 1412 T^{4} - 74 p^{4} T^{5} + 19 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 20 T - 18 p^{2} T^{2} - 1680 T^{3} + 4342123 T^{4} - 1680 p^{3} T^{5} - 18 p^{8} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 + 108 T + 7126 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 180 T + 16130 T^{2} - 1159920 T^{3} + 81394131 T^{4} - 1159920 p^{3} T^{5} + 16130 p^{6} T^{6} - 180 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 64 T - 10462 T^{2} + 53760 T^{3} + 136795019 T^{4} + 53760 p^{3} T^{5} - 10462 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 22 T - 10677 T^{2} - 289806 T^{3} - 29356796 T^{4} - 289806 p^{3} T^{5} - 10677 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 478 T + 102955 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 540 T + 162062 T^{2} + 37776240 T^{3} + 7205534163 T^{4} + 37776240 p^{3} T^{5} + 162062 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 176 T - 1394 p T^{2} + 3300352 T^{3} + 2680409179 T^{4} + 3300352 p^{3} T^{5} - 1394 p^{7} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 46 T - 16373 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 254 T + 164793 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 508 T + 21966 T^{2} - 14453616 T^{3} + 18170071603 T^{4} - 14453616 p^{3} T^{5} + 21966 p^{6} T^{6} - 508 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 344 T - 111666 T^{2} + 23306688 T^{3} + 13119050203 T^{4} + 23306688 p^{3} T^{5} - 111666 p^{6} T^{6} - 344 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 184 T - 332190 T^{2} + 8227008 T^{3} + 84855829051 T^{4} + 8227008 p^{3} T^{5} - 332190 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 94 T + 69521 T^{2} - 48376818 T^{3} - 49843309252 T^{4} - 48376818 p^{3} T^{5} + 69521 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 710 T - 52285 T^{2} - 32050110 T^{3} + 67491257756 T^{4} - 32050110 p^{3} T^{5} - 52285 p^{6} T^{6} + 710 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 1368 T + 1049542 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 76 T - 2974 p T^{2} - 42191856 T^{3} - 103906585597 T^{4} - 42191856 p^{3} T^{5} - 2974 p^{7} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 24 T - 8170 p T^{2} - 8161728 T^{3} + 173952306051 T^{4} - 8161728 p^{3} T^{5} - 8170 p^{7} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 142 T + 334921 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 602 T - 872439 T^{2} + 105407190 T^{3} + 772372358212 T^{4} + 105407190 p^{3} T^{5} - 872439 p^{6} T^{6} - 602 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 + 1420 T + 2163846 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.90276013843510378271700173110, −6.77308763724741481770067559782, −6.21352225505920134512791979305, −5.97239997311785610490838011310, −5.92506511184764099625286214851, −5.51258709319841630244180836750, −5.21262645881463647578895036302, −5.19677034950922066861664281250, −5.00635450224091266373253406507, −4.78112770415109879026025238908, −4.69533588101848731266871824461, −4.35496233637070349732656240926, −4.08275803808906892457851782707, −3.63796258578918160926788116576, −3.18270841953286883555317589009, −3.02856123359015804552195684788, −2.64176217346433953466344146134, −2.58220407079263874325723763971, −2.44610266656211069862529574904, −1.94079638605940203905253851661, −1.65061663242400028951526461267, −1.19935366120057728306365783763, −0.65233948065699361462010658506, −0.54988543799359640985391912736, −0.45442676672863313064604040874, 0.45442676672863313064604040874, 0.54988543799359640985391912736, 0.65233948065699361462010658506, 1.19935366120057728306365783763, 1.65061663242400028951526461267, 1.94079638605940203905253851661, 2.44610266656211069862529574904, 2.58220407079263874325723763971, 2.64176217346433953466344146134, 3.02856123359015804552195684788, 3.18270841953286883555317589009, 3.63796258578918160926788116576, 4.08275803808906892457851782707, 4.35496233637070349732656240926, 4.69533588101848731266871824461, 4.78112770415109879026025238908, 5.00635450224091266373253406507, 5.19677034950922066861664281250, 5.21262645881463647578895036302, 5.51258709319841630244180836750, 5.92506511184764099625286214851, 5.97239997311785610490838011310, 6.21352225505920134512791979305, 6.77308763724741481770067559782, 6.90276013843510378271700173110

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