# Properties

 Label 8-1840e4-1.1-c3e4-0-0 Degree $8$ Conductor $1.146\times 10^{13}$ Sign $1$ Analytic cond. $1.38910\times 10^{8}$ Root an. cond. $10.4193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 20·5-s + 7-s − 30·9-s + 39·11-s − 20·13-s − 80·15-s − 23·17-s − 53·19-s + 4·21-s + 92·23-s + 250·25-s − 223·27-s + 161·29-s − 388·31-s + 156·33-s − 20·35-s + 466·37-s − 80·39-s + 484·41-s − 894·43-s + 600·45-s + 265·47-s + 136·49-s − 92·51-s + 576·53-s − 780·55-s + ⋯
 L(s)  = 1 + 0.769·3-s − 1.78·5-s + 0.0539·7-s − 1.11·9-s + 1.06·11-s − 0.426·13-s − 1.37·15-s − 0.328·17-s − 0.639·19-s + 0.0415·21-s + 0.834·23-s + 2·25-s − 1.58·27-s + 1.03·29-s − 2.24·31-s + 0.822·33-s − 0.0965·35-s + 2.07·37-s − 0.328·39-s + 1.84·41-s − 3.17·43-s + 1.98·45-s + 0.822·47-s + 0.396·49-s − 0.252·51-s + 1.49·53-s − 1.91·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{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 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 5^{4} \cdot 23^{4}$$ Sign: $1$ Analytic conductor: $$1.38910\times 10^{8}$$ Root analytic conductor: $$10.4193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$7.417339399$$ $$L(\frac12)$$ $$\approx$$ $$7.417339399$$ $$L(\frac{5}{2})$$ 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$C_1$ $$( 1 + p T )^{4}$$
23$C_1$ $$( 1 - p T )^{4}$$
good3$C_2 \wr S_4$ $$1 - 4 T + 46 T^{2} - p^{4} T^{3} + 1190 T^{4} - p^{7} T^{5} + 46 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8}$$
7$C_2 \wr S_4$ $$1 - T - 135 T^{2} - 521 p T^{3} + 125756 T^{4} - 521 p^{4} T^{5} - 135 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8}$$
11$C_2 \wr S_4$ $$1 - 39 T + 323 p T^{2} - 61307 T^{3} + 4937428 T^{4} - 61307 p^{3} T^{5} + 323 p^{7} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr S_4$ $$1 + 20 T + 3840 T^{2} + 157301 T^{3} + 10278344 T^{4} + 157301 p^{3} T^{5} + 3840 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr S_4$ $$1 + 23 T + 5447 T^{2} - 251497 T^{3} + 6295268 T^{4} - 251497 p^{3} T^{5} + 5447 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 + 53 T + 653 p T^{2} + 750501 T^{3} + 76029128 T^{4} + 750501 p^{3} T^{5} + 653 p^{7} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 - 161 T + 2424 p T^{2} - 9893175 T^{3} + 2240848710 T^{4} - 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} - 161 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 + 388 T + 132240 T^{2} + 29130015 T^{3} + 5707089166 T^{4} + 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} + 388 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - 466 T + 240532 T^{2} - 71494750 T^{3} + 19227960950 T^{4} - 71494750 p^{3} T^{5} + 240532 p^{6} T^{6} - 466 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 + 894 T + 484780 T^{2} + 190414350 T^{3} + 59052057270 T^{4} + 190414350 p^{3} T^{5} + 484780 p^{6} T^{6} + 894 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 - 265 T + 184902 T^{2} - 39015297 T^{3} + 18141867922 T^{4} - 39015297 p^{3} T^{5} + 184902 p^{6} T^{6} - 265 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 576 T + 488156 T^{2} - 177483648 T^{3} + 97785187798 T^{4} - 177483648 p^{3} T^{5} + 488156 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 - 94 T + 578056 T^{2} - 37319902 T^{3} + 164753482782 T^{4} - 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 - 1472 T + 1805084 T^{2} - 1370871744 T^{3} + 894579577654 T^{4} - 1370871744 p^{3} T^{5} + 1805084 p^{6} T^{6} - 1472 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 + 200 T + 364322 T^{2} + 121468265 T^{3} + 278790401066 T^{4} + 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 - 1147 T + 1912188 T^{2} - 1347657401 T^{3} + 1180324718270 T^{4} - 1347657401 p^{3} T^{5} + 1912188 p^{6} T^{6} - 1147 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 - 908 T + 1048492 T^{2} - 1070426012 T^{3} + 693505308902 T^{4} - 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} - 908 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 - 1048 T + 1927704 T^{2} - 1122552888 T^{3} + 1371917773086 T^{4} - 1122552888 p^{3} T^{5} + 1927704 p^{6} T^{6} - 1048 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 + 1784 T + 2327792 T^{2} + 1523808488 T^{3} + 1318662054974 T^{4} + 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} + 1784 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 + 2047 T + 2933893 T^{2} + 2857813609 T^{3} + 3031159147844 T^{4} + 2857813609 p^{3} T^{5} + 2933893 p^{6} T^{6} + 2047 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$