# Properties

 Label 8-6762e4-1.1-c1e4-0-6 Degree $8$ Conductor $2.091\times 10^{15}$ Sign $1$ Analytic cond. $8.49980\times 10^{6}$ Root an. cond. $7.34811$ 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 + 6·5-s − 16·6-s + 20·8-s + 10·9-s + 24·10-s − 40·12-s + 10·13-s − 24·15-s + 35·16-s + 12·17-s + 40·18-s + 8·19-s + 60·20-s − 4·23-s − 80·24-s + 11·25-s + 40·26-s − 20·27-s + 6·29-s − 96·30-s + 8·31-s + 56·32-s + 48·34-s + 100·36-s + ⋯
 L(s)  = 1 + 2.82·2-s − 2.30·3-s + 5·4-s + 2.68·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s + 7.58·10-s − 11.5·12-s + 2.77·13-s − 6.19·15-s + 35/4·16-s + 2.91·17-s + 9.42·18-s + 1.83·19-s + 13.4·20-s − 0.834·23-s − 16.3·24-s + 11/5·25-s + 7.84·26-s − 3.84·27-s + 1.11·29-s − 17.5·30-s + 1.43·31-s + 9.89·32-s + 8.23·34-s + 50/3·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{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 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}$$ Sign: $1$ Analytic conductor: $$8.49980\times 10^{6}$$ Root analytic conductor: $$7.34811$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{6762} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$125.0575316$$ $$L(\frac12)$$ $$\approx$$ $$125.0575316$$ $$L(\frac{3}{2})$$ 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 $$1$$
23$C_1$ $$( 1 + T )^{4}$$
good5$C_2 \wr C_2\wr C_2$ $$1 - 6 T + p^{2} T^{2} - 78 T^{3} + 202 T^{4} - 78 p T^{5} + p^{4} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 26 T^{2} + 8 T^{3} + 386 T^{4} + 8 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 88 T^{2} - 516 T^{3} + 2446 T^{4} - 516 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 6 T + 103 T^{2} - 390 T^{3} + 4096 T^{4} - 390 p T^{5} + 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
31$D_{4}$ $$( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 6 T + 127 T^{2} + 622 T^{3} + 6656 T^{4} + 622 p T^{5} + 127 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 14 T + 195 T^{2} - 1734 T^{3} + 12626 T^{4} - 1734 p T^{5} + 195 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 6 T + 125 T^{2} + 606 T^{3} + 7708 T^{4} + 606 p T^{5} + 125 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 165 T^{2} - 242 T^{3} + 11090 T^{4} - 242 p T^{5} + 165 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 134 T^{2} + 164 T^{3} + 8002 T^{4} + 164 p T^{5} + 134 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 218 T^{2} - 1240 T^{3} + 18538 T^{4} - 1240 p T^{5} + 218 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 212 T^{2} - 2140 T^{3} + 18546 T^{4} - 2140 p T^{5} + 212 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 224 T^{2} + 772 T^{3} + 21102 T^{4} + 772 p T^{5} + 224 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 266 T^{2} - 8 T^{3} + 27746 T^{4} - 8 p T^{5} + 266 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 276 T^{2} - 2316 T^{3} + 402 p T^{4} - 2316 p T^{5} + 276 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 24 T + 514 T^{2} + 6328 T^{3} + 69386 T^{4} + 6328 p T^{5} + 514 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 330 T^{2} - 3528 T^{3} + 40994 T^{4} - 3528 p T^{5} + 330 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 262 T^{2} - 2076 T^{3} + 31042 T^{4} - 2076 p T^{5} + 262 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 30 T + 601 T^{2} - 8294 T^{3} + 91892 T^{4} - 8294 p T^{5} + 601 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$