# Properties

 Label 8-3822e4-1.1-c1e4-0-4 Degree $8$ Conductor $2.134\times 10^{14}$ Sign $1$ Analytic cond. $867503.$ Root an. cond. $5.52438$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s − 4·3-s + 10·4-s + 2·5-s − 16·6-s + 20·8-s + 10·9-s + 8·10-s + 6·11-s − 40·12-s − 4·13-s − 8·15-s + 35·16-s − 2·17-s + 40·18-s + 2·19-s + 20·20-s + 24·22-s + 10·23-s − 80·24-s − 5·25-s − 16·26-s − 20·27-s + 10·29-s − 32·30-s + 56·32-s − 24·33-s + ⋯
 L(s)  = 1 + 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s + 2.52·10-s + 1.80·11-s − 11.5·12-s − 1.10·13-s − 2.06·15-s + 35/4·16-s − 0.485·17-s + 9.42·18-s + 0.458·19-s + 4.47·20-s + 5.11·22-s + 2.08·23-s − 16.3·24-s − 25-s − 3.13·26-s − 3.84·27-s + 1.85·29-s − 5.84·30-s + 9.89·32-s − 4.17·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{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^{4} \cdot 3^{4} \cdot 7^{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: $$2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$867503.$$ Root analytic conductor: $$5.52438$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$45.75292696$$ $$L(\frac12)$$ $$\approx$$ $$45.75292696$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{4}$$
3$C_1$ $$( 1 + T )^{4}$$
7 $$1$$
13$C_1$ $$( 1 + T )^{4}$$
good5$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 9 T^{2} - 18 T^{3} + 68 T^{4} - 18 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 - 6 T + 35 T^{2} - 118 T^{3} + 464 T^{4} - 118 p T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 47 T^{2} + 106 T^{3} + 1048 T^{4} + 106 p T^{5} + 47 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 65 T^{2} - 102 T^{3} + 1776 T^{4} - 102 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 117 T^{2} - 30 p T^{3} + 4316 T^{4} - 30 p^{2} T^{5} + 117 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 123 T^{2} - 858 T^{3} + 5450 T^{4} - 858 p T^{5} + 123 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 + 98 T^{2} + 16 T^{3} + 4266 T^{4} + 16 p T^{5} + 98 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 - 6 T + 113 T^{2} - 486 T^{3} + 5578 T^{4} - 486 p T^{5} + 113 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 138 T^{2} - 16 T^{3} + 8066 T^{4} - 16 p T^{5} + 138 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 125 T^{2} - 986 T^{3} + 8068 T^{4} - 986 p T^{5} + 125 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 162 T^{2} - 880 T^{3} + 222 p T^{4} - 880 p T^{5} + 162 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 274 T^{2} - 2576 T^{3} + 23362 T^{4} - 2576 p T^{5} + 274 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 210 T^{2} + 16 T^{3} + 17930 T^{4} + 16 p T^{5} + 210 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 167 T^{2} + 658 T^{3} + 12616 T^{4} + 658 p T^{5} + 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 28 T + 508 T^{2} - 6308 T^{3} + 59698 T^{4} - 6308 p T^{5} + 508 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 218 T^{2} - 2424 T^{3} + 24770 T^{4} - 2424 p T^{5} + 218 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 14 T + 353 T^{2} - 3150 T^{3} + 40908 T^{4} - 3150 p T^{5} + 353 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 234 T^{2} - 1360 T^{3} + 24546 T^{4} - 1360 p T^{5} + 234 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 224 T^{2} + 516 T^{3} + 22958 T^{4} + 516 p T^{5} + 224 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 260 T^{2} + 452 T^{3} + 29794 T^{4} + 452 p T^{5} + 260 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 304 T^{2} + 1196 T^{3} + 40606 T^{4} + 1196 p T^{5} + 304 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$