Properties

Label 8-1080e4-1.1-c3e4-0-0
Degree $8$
Conductor $1.360\times 10^{12}$
Sign $1$
Analytic cond. $1.64876\times 10^{7}$
Root an. cond. $7.98261$
Motivic weight $3$
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  − 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

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.64876\times 10^{7}\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1080} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.857148734\)
\(L(\frac12)\) \(\approx\) \(6.857148734\)
\(L(\frac{5}{2})\) 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

−6.79649452922043283608233435485, −6.34361887533998246677225635459, −6.26888605426022496019276876647, −6.02535721891704731312272546076, −5.88653586428700622052503217595, −5.30215623459964804362932129497, −5.26661692363285108499071173637, −5.09186055269416203694297526297, −4.96251707276494288359305420012, −4.54336358652557692552680739202, −4.21183733859880638453836429849, −4.04983334776332151277510175782, −3.92423676341352005899805972166, −3.53161479933345638993711836881, −3.42763694483596344429979109832, −3.27218381271347477918585687449, −2.99769819163523705684466175790, −2.24531282170143901190608403049, −2.14059392886521308445704153113, −2.07815457721748687394197852869, −1.63876580929731679716332915753, −1.13369822195653223148535263466, −0.65516536917955004113726460931, −0.58136826099589294356913303943, −0.49664902847978265944941862643, 0.49664902847978265944941862643, 0.58136826099589294356913303943, 0.65516536917955004113726460931, 1.13369822195653223148535263466, 1.63876580929731679716332915753, 2.07815457721748687394197852869, 2.14059392886521308445704153113, 2.24531282170143901190608403049, 2.99769819163523705684466175790, 3.27218381271347477918585687449, 3.42763694483596344429979109832, 3.53161479933345638993711836881, 3.92423676341352005899805972166, 4.04983334776332151277510175782, 4.21183733859880638453836429849, 4.54336358652557692552680739202, 4.96251707276494288359305420012, 5.09186055269416203694297526297, 5.26661692363285108499071173637, 5.30215623459964804362932129497, 5.88653586428700622052503217595, 6.02535721891704731312272546076, 6.26888605426022496019276876647, 6.34361887533998246677225635459, 6.79649452922043283608233435485

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