# Properties

 Label 8-9680e4-1.1-c1e4-0-5 Degree $8$ Conductor $8.780\times 10^{15}$ Sign $1$ Analytic cond. $3.56952\times 10^{7}$ Root an. cond. $8.79176$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 4·5-s + 7·7-s − 6·9-s + 3·13-s + 8·15-s + 11·17-s − 2·19-s + 14·21-s + 11·23-s + 10·25-s − 17·27-s + 4·29-s + 17·31-s + 28·35-s − 3·37-s + 6·39-s + 13·41-s + 7·43-s − 24·45-s + 47-s + 8·49-s + 22·51-s − 15·53-s − 4·57-s + 17·59-s + 4·61-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.78·5-s + 2.64·7-s − 2·9-s + 0.832·13-s + 2.06·15-s + 2.66·17-s − 0.458·19-s + 3.05·21-s + 2.29·23-s + 2·25-s − 3.27·27-s + 0.742·29-s + 3.05·31-s + 4.73·35-s − 0.493·37-s + 0.960·39-s + 2.03·41-s + 1.06·43-s − 3.57·45-s + 0.145·47-s + 8/7·49-s + 3.08·51-s − 2.06·53-s − 0.529·57-s + 2.21·59-s + 0.512·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 5^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$3.56952\times 10^{7}$$ Root analytic conductor: $$8.79176$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$62.23431547$$ $$L(\frac12)$$ $$\approx$$ $$62.23431547$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 - T )^{4}$$
11 $$1$$
good3$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 10 T^{2} - 5 p T^{3} + 43 T^{4} - 5 p^{2} T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
7$C_2 \wr C_2\wr C_2$ $$1 - p T + 41 T^{2} - 22 p T^{3} + 477 T^{4} - 22 p^{2} T^{5} + 41 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - 3 T + 31 T^{2} - 8 p T^{3} + 509 T^{4} - 8 p^{2} T^{5} + 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 11 T + 88 T^{2} - 499 T^{3} + 2403 T^{4} - 499 p T^{5} + 88 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 34 T^{2} + 141 T^{3} + 671 T^{4} + 141 p T^{5} + 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 11 T + 134 T^{2} - 823 T^{3} + 5137 T^{4} - 823 p T^{5} + 134 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 68 T^{2} - 379 T^{3} + 2263 T^{4} - 379 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 17 T + 187 T^{2} - 1518 T^{3} + 9293 T^{4} - 1518 p T^{5} + 187 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 3 T + 48 T^{2} + 309 T^{3} + 813 T^{4} + 309 p T^{5} + 48 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 13 T + 158 T^{2} - 1301 T^{3} + 10135 T^{4} - 1301 p T^{5} + 158 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 7 T + 155 T^{2} - 810 T^{3} + 9693 T^{4} - 810 p T^{5} + 155 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - T + 125 T^{2} + 100 T^{3} + 7183 T^{4} + 100 p T^{5} + 125 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 15 T + 201 T^{2} + 1640 T^{3} + 14327 T^{4} + 1640 p T^{5} + 201 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 17 T + 201 T^{2} - 1738 T^{3} + 15075 T^{4} - 1738 p T^{5} + 201 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 111 T^{2} - 638 T^{3} + 6911 T^{4} - 638 p T^{5} + 111 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 7 T + 176 T^{2} + 1169 T^{3} + 15037 T^{4} + 1169 p T^{5} + 176 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 15 T + 140 T^{2} - 385 T^{3} - 73 T^{4} - 385 p T^{5} + 140 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 7 T + 2 p T^{2} - 901 T^{3} + 14409 T^{4} - 901 p T^{5} + 2 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 244 T^{2} - 2161 T^{3} + 28451 T^{4} - 2161 p T^{5} + 244 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 9 T + 359 T^{2} + 2272 T^{3} + 45827 T^{4} + 2272 p T^{5} + 359 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 12 T + 274 T^{2} + 24 p T^{3} + 32451 T^{4} + 24 p^{2} T^{5} + 274 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 288 T^{2} - 436 T^{3} + 39233 T^{4} - 436 p T^{5} + 288 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$