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Properties

Label 8-2160e4-1.1-c3e4-0-3
Degree $8$
Conductor $2.177\times 10^{13}$
Sign $1$
Analytic cond. $2.63802\times 10^{8}$
Root an. cond. $11.2891$
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  − 20·5-s − 14·7-s − 4·11-s + 30·13-s + 28·17-s − 78·19-s + 182·23-s + 250·25-s − 202·29-s + 76·31-s + 280·35-s + 302·37-s − 380·41-s − 178·43-s + 114·47-s − 109·49-s + 256·53-s + 80·55-s − 204·59-s + 766·61-s − 600·65-s − 330·67-s − 1.06e3·71-s + 1.44e3·73-s + 56·77-s − 742·79-s − 768·83-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.755·7-s − 0.109·11-s + 0.640·13-s + 0.399·17-s − 0.941·19-s + 1.64·23-s + 2·25-s − 1.29·29-s + 0.440·31-s + 1.35·35-s + 1.34·37-s − 1.44·41-s − 0.631·43-s + 0.353·47-s − 0.317·49-s + 0.663·53-s + 0.196·55-s − 0.450·59-s + 1.60·61-s − 1.14·65-s − 0.601·67-s − 1.77·71-s + 2.31·73-s + 0.0828·77-s − 1.05·79-s − 1.01·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.63802\times 10^{8}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 2 p T + 305 T^{2} + 4614 T^{3} + 488 p T^{4} + 4614 p^{3} T^{5} + 305 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4 T + 1575 T^{2} - 29784 T^{3} + 2457124 T^{4} - 29784 p^{3} T^{5} + 1575 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 30 T - 140 T^{2} - 26820 T^{3} + 7961973 T^{4} - 26820 p^{3} T^{5} - 140 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 28 T + 6099 T^{2} - 159456 T^{3} + 948020 p T^{4} - 159456 p^{3} T^{5} + 6099 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 78 T + 16433 T^{2} + 1084170 T^{3} + 152743716 T^{4} + 1084170 p^{3} T^{5} + 16433 p^{6} T^{6} + 78 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 182 T + 33285 T^{2} - 2652138 T^{3} + 367341256 T^{4} - 2652138 p^{3} T^{5} + 33285 p^{6} T^{6} - 182 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 202 T + 54117 T^{2} + 3130494 T^{3} + 888383644 T^{4} + 3130494 p^{3} T^{5} + 54117 p^{6} T^{6} + 202 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 76 T + 52175 T^{2} - 3793644 T^{3} + 2235420704 T^{4} - 3793644 p^{3} T^{5} + 52175 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 302 T + 144869 T^{2} - 1237962 p T^{3} + 9554438060 T^{4} - 1237962 p^{4} T^{5} + 144869 p^{6} T^{6} - 302 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 380 T + 304944 T^{2} + 77291940 T^{3} + 32535322366 T^{4} + 77291940 p^{3} T^{5} + 304944 p^{6} T^{6} + 380 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 178 T + 211805 T^{2} + 53964162 T^{3} + 20664465836 T^{4} + 53964162 p^{3} T^{5} + 211805 p^{6} T^{6} + 178 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 114 T + 308433 T^{2} - 39725310 T^{3} + 42951666476 T^{4} - 39725310 p^{3} T^{5} + 308433 p^{6} T^{6} - 114 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 256 T + 177264 T^{2} - 81601152 T^{3} + 34810114894 T^{4} - 81601152 p^{3} T^{5} + 177264 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 204 T + 83084 T^{2} - 95352084 T^{3} - 46391911626 T^{4} - 95352084 p^{3} T^{5} + 83084 p^{6} T^{6} + 204 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 766 T + 444473 T^{2} - 98451618 T^{3} + 48468291764 T^{4} - 98451618 p^{3} T^{5} + 444473 p^{6} T^{6} - 766 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 330 T + 1123349 T^{2} + 294102570 T^{3} + 495222287532 T^{4} + 294102570 p^{3} T^{5} + 1123349 p^{6} T^{6} + 330 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1060 T + 1430856 T^{2} + 1119382500 T^{3} + 767140525006 T^{4} + 1119382500 p^{3} T^{5} + 1430856 p^{6} T^{6} + 1060 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1442 T + 1995917 T^{2} - 1634684910 T^{3} + 1251343303316 T^{4} - 1634684910 p^{3} T^{5} + 1995917 p^{6} T^{6} - 1442 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 742 T + 759772 T^{2} + 600002584 T^{3} + 627086485909 T^{4} + 600002584 p^{3} T^{5} + 759772 p^{6} T^{6} + 742 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 768 T + 2075540 T^{2} + 1295690112 T^{3} + 1724588712726 T^{4} + 1295690112 p^{3} T^{5} + 2075540 p^{6} T^{6} + 768 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 400 T + 2456228 T^{2} - 740686960 T^{3} + 2486053553638 T^{4} - 740686960 p^{3} T^{5} + 2456228 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 3338 T + 6353893 T^{2} - 8727311702 T^{3} + 9281975879860 T^{4} - 8727311702 p^{3} T^{5} + 6353893 p^{6} T^{6} - 3338 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.67013207047215842814041817779, −6.28437737909305034887817509677, −6.02754366729099058801556211595, −6.02416231020477782493989657313, −5.73537117788599248122141209537, −5.32430872511825248893531437693, −5.15748677329458019868069892075, −4.96926929817038081508935351691, −4.89345235458928274085556640474, −4.56304193048878833325781966818, −4.34088185540242154088978749853, −3.98172740002470697049815080978, −3.95703409423345648137826620107, −3.66188762588000779099928579359, −3.37022808491706242054982126823, −3.33854041190009849974516095342, −3.28231741178723094787648609750, −2.64057127089699552993416121136, −2.45109460755093593343389066785, −2.29964731430367246191271985723, −2.25668718417237057877709237878, −1.29226625441494257434469020694, −1.25009991303512396055323738024, −1.13885379776093629722219097671, −1.00013694736494124864172200017, 0, 0, 0, 0, 1.00013694736494124864172200017, 1.13885379776093629722219097671, 1.25009991303512396055323738024, 1.29226625441494257434469020694, 2.25668718417237057877709237878, 2.29964731430367246191271985723, 2.45109460755093593343389066785, 2.64057127089699552993416121136, 3.28231741178723094787648609750, 3.33854041190009849974516095342, 3.37022808491706242054982126823, 3.66188762588000779099928579359, 3.95703409423345648137826620107, 3.98172740002470697049815080978, 4.34088185540242154088978749853, 4.56304193048878833325781966818, 4.89345235458928274085556640474, 4.96926929817038081508935351691, 5.15748677329458019868069892075, 5.32430872511825248893531437693, 5.73537117788599248122141209537, 6.02416231020477782493989657313, 6.02754366729099058801556211595, 6.28437737909305034887817509677, 6.67013207047215842814041817779

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