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Properties

Label 8-1083e4-1.1-c3e4-0-2
Degree $8$
Conductor $1.376\times 10^{12}$
Sign $1$
Analytic cond. $1.66716\times 10^{7}$
Root an. cond. $7.99368$
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  − 2·2-s + 12·3-s − 3·4-s + 18·5-s − 24·6-s − 10·7-s − 4·8-s + 90·9-s − 36·10-s − 14·11-s − 36·12-s − 56·13-s + 20·14-s + 216·15-s + 39·16-s − 186·17-s − 180·18-s − 54·20-s − 120·21-s + 28·22-s − 84·23-s − 48·24-s − 173·25-s + 112·26-s + 540·27-s + 30·28-s + 236·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.30·3-s − 3/8·4-s + 1.60·5-s − 1.63·6-s − 0.539·7-s − 0.176·8-s + 10/3·9-s − 1.13·10-s − 0.383·11-s − 0.866·12-s − 1.19·13-s + 0.381·14-s + 3.71·15-s + 0.609·16-s − 2.65·17-s − 2.35·18-s − 0.603·20-s − 1.24·21-s + 0.271·22-s − 0.761·23-s − 0.408·24-s − 1.38·25-s + 0.844·26-s + 3.84·27-s + 0.202·28-s + 1.51·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(1.66716\times 10^{7}\)
Root analytic conductor: \(7.99368\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 19^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 - p T )^{4} \)
19 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + p T + 7 T^{2} + 3 p^{3} T^{3} + 19 p T^{4} + 3 p^{6} T^{5} + 7 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 18 T + 497 T^{2} - 5634 T^{3} + 89156 T^{4} - 5634 p^{3} T^{5} + 497 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 10 T + 957 T^{2} + 11898 T^{3} + 419908 T^{4} + 11898 p^{3} T^{5} + 957 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 14 T + 2049 T^{2} + 16954 T^{3} + 3844088 T^{4} + 16954 p^{3} T^{5} + 2049 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 56 T + 7388 T^{2} + 271368 T^{3} + 22128422 T^{4} + 271368 p^{3} T^{5} + 7388 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 186 T + 89 p^{2} T^{2} + 2469834 T^{3} + 193997780 T^{4} + 2469834 p^{3} T^{5} + 89 p^{8} T^{6} + 186 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 84 T + 30344 T^{2} + 1222788 T^{3} + 17501794 p T^{4} + 1222788 p^{3} T^{5} + 30344 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 236 T + 38296 T^{2} - 2172756 T^{3} + 400254782 T^{4} - 2172756 p^{3} T^{5} + 38296 p^{6} T^{6} - 236 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 112 T + 76212 T^{2} + 6302064 T^{3} + 2836990870 T^{4} + 6302064 p^{3} T^{5} + 76212 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 544 T + 224364 T^{2} + 1677600 p T^{3} + 16007180134 T^{4} + 1677600 p^{4} T^{5} + 224364 p^{6} T^{6} + 544 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 68 T + 159832 T^{2} - 24974988 T^{3} + 12851495438 T^{4} - 24974988 p^{3} T^{5} + 159832 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 674 T + 355397 T^{2} + 126100458 T^{3} + 39633391484 T^{4} + 126100458 p^{3} T^{5} + 355397 p^{6} T^{6} + 674 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 114 T + 171337 T^{2} - 6705570 T^{3} + 18249717032 T^{4} - 6705570 p^{3} T^{5} + 171337 p^{6} T^{6} + 114 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 540 T + 372744 T^{2} + 73265892 T^{3} + 44247102910 T^{4} + 73265892 p^{3} T^{5} + 372744 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 582764 T^{2} + 677376 T^{3} + 165505100406 T^{4} + 677376 p^{3} T^{5} + 582764 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 558 T + 87745 T^{2} - 60123114 T^{3} - 12563228676 T^{4} - 60123114 p^{3} T^{5} + 87745 p^{6} T^{6} + 558 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 320 T + 567788 T^{2} - 76531008 T^{3} + 195546355478 T^{4} - 76531008 p^{3} T^{5} + 567788 p^{6} T^{6} - 320 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 872 T + 1195948 T^{2} + 855329544 T^{3} + 617068843718 T^{4} + 855329544 p^{3} T^{5} + 1195948 p^{6} T^{6} + 872 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 446 T + 949913 T^{2} - 497765814 T^{3} + 450278879396 T^{4} - 497765814 p^{3} T^{5} + 949913 p^{6} T^{6} - 446 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1232 T + 1638900 T^{2} - 1309543440 T^{3} + 1185149213782 T^{4} - 1309543440 p^{3} T^{5} + 1638900 p^{6} T^{6} - 1232 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 780 T + 1378672 T^{2} + 750270540 T^{3} + 1034898364334 T^{4} + 750270540 p^{3} T^{5} + 1378672 p^{6} T^{6} + 780 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 60 T + 670872 T^{2} + 799448436 T^{3} - 22570573490 T^{4} + 799448436 p^{3} T^{5} + 670872 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2224 T + 3918308 T^{2} - 4720818960 T^{3} + 5246860342598 T^{4} - 4720818960 p^{3} T^{5} + 3918308 p^{6} T^{6} - 2224 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.99444767617302964160401139723, −6.78880942880939623954790292796, −6.78544317402228197397232799293, −6.47369784806651124658929465325, −6.36346002547948805656316049875, −6.34842099289411731514810344484, −5.61852994564036756428898907069, −5.57216143999279570689579074591, −5.30308179764151773959137744652, −5.06933111729948751757873268485, −4.71292160729323287850402362016, −4.65936687596261913338138112576, −4.30230345622050608631866897412, −3.98013480037129183245341730098, −3.69339932173430345584360085638, −3.62542530743543034024802280781, −3.16182031585906213137865382418, −2.95894232225769712613865729955, −2.78833382287160930987867901153, −2.23475035145670986801937773515, −2.22194112583618831754088508531, −2.16145006169813327280591812159, −1.58843449863729883963914821559, −1.41068531544588663315762371292, −1.39488154571977503729677050011, 0, 0, 0, 0, 1.39488154571977503729677050011, 1.41068531544588663315762371292, 1.58843449863729883963914821559, 2.16145006169813327280591812159, 2.22194112583618831754088508531, 2.23475035145670986801937773515, 2.78833382287160930987867901153, 2.95894232225769712613865729955, 3.16182031585906213137865382418, 3.62542530743543034024802280781, 3.69339932173430345584360085638, 3.98013480037129183245341730098, 4.30230345622050608631866897412, 4.65936687596261913338138112576, 4.71292160729323287850402362016, 5.06933111729948751757873268485, 5.30308179764151773959137744652, 5.57216143999279570689579074591, 5.61852994564036756428898907069, 6.34842099289411731514810344484, 6.36346002547948805656316049875, 6.47369784806651124658929465325, 6.78544317402228197397232799293, 6.78880942880939623954790292796, 6.99444767617302964160401139723

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