# Properties

 Label 8-3072e4-1.1-c1e4-0-12 Degree $8$ Conductor $8.906\times 10^{13}$ Sign $1$ Analytic cond. $362070.$ Root an. cond. $4.95278$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 4·5-s + 4·7-s + 10·9-s − 8·13-s + 16·15-s − 16·21-s − 20·27-s − 12·29-s + 12·31-s − 16·35-s − 16·37-s + 32·39-s − 40·45-s − 4·49-s − 20·53-s − 16·61-s + 40·63-s + 32·65-s + 16·67-s − 8·71-s − 8·73-s + 12·79-s + 35·81-s + 48·87-s − 8·89-s − 32·91-s + ⋯
 L(s)  = 1 − 2.30·3-s − 1.78·5-s + 1.51·7-s + 10/3·9-s − 2.21·13-s + 4.13·15-s − 3.49·21-s − 3.84·27-s − 2.22·29-s + 2.15·31-s − 2.70·35-s − 2.63·37-s + 5.12·39-s − 5.96·45-s − 4/7·49-s − 2.74·53-s − 2.04·61-s + 5.03·63-s + 3.96·65-s + 1.95·67-s − 0.949·71-s − 0.936·73-s + 1.35·79-s + 35/9·81-s + 5.14·87-s − 0.847·89-s − 3.35·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{40} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$362070.$$ Root analytic conductor: $$4.95278$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3072} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{4}$$
good5$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
7$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 20 T^{2} - 60 T^{3} + 186 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 20 T^{2} - 32 T^{3} + 230 T^{4} - 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 56 T^{2} + 264 T^{3} + 1122 T^{4} + 264 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + 38 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2 \wr C_2\wr C_2$ $$1 + 12 T + 144 T^{2} + 964 T^{3} + 6422 T^{4} + 964 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 16 T + 200 T^{2} + 1552 T^{3} + 11010 T^{4} + 1552 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 100 T^{2} + 192 T^{3} + 4726 T^{4} + 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 20 T + 336 T^{2} + 3452 T^{3} + 30134 T^{4} + 3452 p T^{5} + 336 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 16 T + 296 T^{2} + 2704 T^{3} + 27618 T^{4} + 2704 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 116 T^{2} - 160 T^{3} + 5510 T^{4} - 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 156 T^{2} + 504 T^{3} + 10022 T^{4} + 504 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$