# Properties

 Label 8-1560e4-1.1-c3e4-0-1 Degree $8$ Conductor $5.922\times 10^{12}$ Sign $1$ Analytic cond. $7.17732\times 10^{7}$ Root an. cond. $9.59390$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·3-s − 20·5-s − 11·7-s + 90·9-s + 35·11-s − 52·13-s + 240·15-s − 9·17-s + 32·19-s + 132·21-s − 149·23-s + 250·25-s − 540·27-s + 168·29-s + 280·31-s − 420·33-s + 220·35-s + 37·37-s + 624·39-s + 247·41-s + 132·43-s − 1.80e3·45-s − 517·49-s + 108·51-s + 247·53-s − 700·55-s − 384·57-s + ⋯
 L(s)  = 1 − 2.30·3-s − 1.78·5-s − 0.593·7-s + 10/3·9-s + 0.959·11-s − 1.10·13-s + 4.13·15-s − 0.128·17-s + 0.386·19-s + 1.37·21-s − 1.35·23-s + 2·25-s − 3.84·27-s + 1.07·29-s + 1.62·31-s − 2.21·33-s + 1.06·35-s + 0.164·37-s + 2.56·39-s + 0.940·41-s + 0.468·43-s − 5.96·45-s − 1.50·49-s + 0.296·51-s + 0.640·53-s − 1.71·55-s − 0.892·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{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^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$7.17732\times 10^{7}$$ Root analytic conductor: $$9.59390$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## 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$C_1$ $$( 1 + p T )^{4}$$
5$C_1$ $$( 1 + p T )^{4}$$
13$C_1$ $$( 1 + p T )^{4}$$
good7$C_2 \wr S_4$ $$1 + 11 T + 638 T^{2} + 6687 T^{3} + 277250 T^{4} + 6687 p^{3} T^{5} + 638 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8}$$
11$C_2 \wr S_4$ $$1 - 35 T + 3490 T^{2} - 851 p^{2} T^{3} + 6649146 T^{4} - 851 p^{5} T^{5} + 3490 p^{6} T^{6} - 35 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr S_4$ $$1 + 9 T + 18846 T^{2} + 116583 T^{3} + 136826114 T^{4} + 116583 p^{3} T^{5} + 18846 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 - 32 T + 10704 T^{2} - 157808 T^{3} + 49128686 T^{4} - 157808 p^{3} T^{5} + 10704 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr S_4$ $$1 + 149 T + 24036 T^{2} + 2829225 T^{3} + 446878614 T^{4} + 2829225 p^{3} T^{5} + 24036 p^{6} T^{6} + 149 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 - 168 T + 63100 T^{2} - 13021496 T^{3} + 65556974 p T^{4} - 13021496 p^{3} T^{5} + 63100 p^{6} T^{6} - 168 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 - 280 T + 102904 T^{2} - 21276568 T^{3} + 4578875278 T^{4} - 21276568 p^{3} T^{5} + 102904 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - p T + 146826 T^{2} - 702159 T^{3} + 9696983914 T^{4} - 702159 p^{3} T^{5} + 146826 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 - 247 T + 283462 T^{2} - 50131281 T^{3} + 29580506258 T^{4} - 50131281 p^{3} T^{5} + 283462 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 - 132 T + 249180 T^{2} - 23649572 T^{3} + 27653627094 T^{4} - 23649572 p^{3} T^{5} + 249180 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 + 171880 T^{2} - 2485664 T^{3} + 14155391086 T^{4} - 2485664 p^{3} T^{5} + 171880 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 247 T + 543922 T^{2} - 97114533 T^{3} + 117321834362 T^{4} - 97114533 p^{3} T^{5} + 543922 p^{6} T^{6} - 247 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 - 614 T + 566732 T^{2} - 138319766 T^{3} + 106867828150 T^{4} - 138319766 p^{3} T^{5} + 566732 p^{6} T^{6} - 614 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 - 719 T + 676314 T^{2} - 315515861 T^{3} + 199831686026 T^{4} - 315515861 p^{3} T^{5} + 676314 p^{6} T^{6} - 719 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 - 658 T + 433188 T^{2} + 115705454 T^{3} - 53997391258 T^{4} + 115705454 p^{3} T^{5} + 433188 p^{6} T^{6} - 658 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 - 939 T + 718494 T^{2} - 519988407 T^{3} + 427957789762 T^{4} - 519988407 p^{3} T^{5} + 718494 p^{6} T^{6} - 939 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 + 1452 T + 1742084 T^{2} + 1433964468 T^{3} + 1026387364694 T^{4} + 1433964468 p^{3} T^{5} + 1742084 p^{6} T^{6} + 1452 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 - 289 T + 1249872 T^{2} - 83356173 T^{3} + 726477052702 T^{4} - 83356173 p^{3} T^{5} + 1249872 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 + 118 T + 14132 T^{2} - 445321898 T^{3} - 52908707834 T^{4} - 445321898 p^{3} T^{5} + 14132 p^{6} T^{6} + 118 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 + 1319 T + 1221066 T^{2} + 69782097 T^{3} - 879945766 p T^{4} + 69782097 p^{3} T^{5} + 1221066 p^{6} T^{6} + 1319 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 + 993 T + 2688746 T^{2} + 1621923903 T^{3} + 3070810482858 T^{4} + 1621923903 p^{3} T^{5} + 2688746 p^{6} T^{6} + 993 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$