# Properties

 Label 8-7800e4-1.1-c1e4-0-4 Degree $8$ Conductor $3.702\times 10^{15}$ Sign $1$ Analytic cond. $1.50482\times 10^{7}$ Root an. cond. $7.89197$ 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 − 7·7-s + 10·9-s + 11-s + 4·13-s − 3·17-s + 28·21-s − 5·23-s − 20·27-s + 8·29-s − 4·33-s − 5·37-s − 16·39-s + 7·41-s − 8·43-s − 10·47-s + 15·49-s + 12·51-s + 5·53-s + 2·59-s + 11·61-s − 70·63-s − 12·67-s + 20·69-s + 15·71-s + 10·73-s − 7·77-s + ⋯
 L(s)  = 1 − 2.30·3-s − 2.64·7-s + 10/3·9-s + 0.301·11-s + 1.10·13-s − 0.727·17-s + 6.11·21-s − 1.04·23-s − 3.84·27-s + 1.48·29-s − 0.696·33-s − 0.821·37-s − 2.56·39-s + 1.09·41-s − 1.21·43-s − 1.45·47-s + 15/7·49-s + 1.68·51-s + 0.686·53-s + 0.260·59-s + 1.40·61-s − 8.81·63-s − 1.46·67-s + 2.40·69-s + 1.78·71-s + 1.17·73-s − 0.797·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$1.50482\times 10^{7}$$ Root analytic conductor: $$7.89197$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{7800} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## 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$C_1$ $$( 1 + T )^{4}$$
5 $$1$$
13$C_1$ $$( 1 - T )^{4}$$
good7$C_2 \wr C_2\wr C_2$ $$1 + p T + 34 T^{2} + 123 T^{3} + 362 T^{4} + 123 p T^{5} + 34 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 - T + 24 T^{2} - 59 T^{3} + 282 T^{4} - 59 p T^{5} + 24 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + 3 T + 42 T^{2} + 117 T^{3} + 858 T^{4} + 117 p T^{5} + 42 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2 \wr C_2\wr C_2$ $$1 + 5 T + 72 T^{2} + 273 T^{3} + 2222 T^{4} + 273 p T^{5} + 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 96 T^{2} - 552 T^{3} + 4142 T^{4} - 552 p T^{5} + 96 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 5 T + 94 T^{2} + 415 T^{3} + 4594 T^{4} + 415 p T^{5} + 94 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 7 T + 156 T^{2} - 839 T^{3} + 9426 T^{4} - 839 p T^{5} + 156 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 152 T^{2} + 888 T^{3} + 9630 T^{4} + 888 p T^{5} + 152 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 + 10 T + 206 T^{2} + 1358 T^{3} + 14818 T^{4} + 1358 p T^{5} + 206 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 5 T + 158 T^{2} - 655 T^{3} + 11506 T^{4} - 655 p T^{5} + 158 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 150 T^{2} - 318 T^{3} + 11810 T^{4} - 318 p T^{5} + 150 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 11 T + 222 T^{2} - 1885 T^{3} + 19674 T^{4} - 1885 p T^{5} + 222 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 12 T + 244 T^{2} + 2188 T^{3} + 23462 T^{4} + 2188 p T^{5} + 244 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 15 T + 254 T^{2} - 2665 T^{3} + 27242 T^{4} - 2665 p T^{5} + 254 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 280 T^{2} - 2022 T^{3} + 30254 T^{4} - 2022 p T^{5} + 280 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + T + 140 T^{2} + 185 T^{3} + 16966 T^{4} + 185 p T^{5} + 140 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 22 T + 342 T^{2} + 3866 T^{3} + 40290 T^{4} + 3866 p T^{5} + 342 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 9 T + 220 T^{2} - 2025 T^{3} + 23586 T^{4} - 2025 p T^{5} + 220 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + T + 116 T^{2} + 599 T^{3} + 10150 T^{4} + 599 p T^{5} + 116 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$