# Properties

 Label 8-6762e4-1.1-c1e4-0-9 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 + 8·5-s + 16·6-s + 20·8-s + 10·9-s + 32·10-s + 4·11-s + 40·12-s + 12·13-s + 32·15-s + 35·16-s − 4·17-s + 40·18-s + 4·19-s + 80·20-s + 16·22-s + 4·23-s + 80·24-s + 24·25-s + 48·26-s + 20·27-s + 8·29-s + 128·30-s − 4·31-s + 56·32-s + ⋯
 L(s)  = 1 + 2.82·2-s + 2.30·3-s + 5·4-s + 3.57·5-s + 6.53·6-s + 7.07·8-s + 10/3·9-s + 10.1·10-s + 1.20·11-s + 11.5·12-s + 3.32·13-s + 8.26·15-s + 35/4·16-s − 0.970·17-s + 9.42·18-s + 0.917·19-s + 17.8·20-s + 3.41·22-s + 0.834·23-s + 16.3·24-s + 24/5·25-s + 9.41·26-s + 3.84·27-s + 1.48·29-s + 23.3·30-s − 0.718·31-s + 9.89·32-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

 $$L(1)$$ $$\approx$$ $$858.6611297$$ $$L(\frac12)$$ $$\approx$$ $$858.6611297$$ $$L(\frac{3}{2})$$ 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$D_{4}$ $$( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 24 T^{2} - 84 T^{3} + 302 T^{4} - 84 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 88 T^{2} - 452 T^{3} + 1838 T^{4} - 452 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 56 T^{2} + 156 T^{3} + 1310 T^{4} + 156 p T^{5} + 56 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 20 T^{2} + 52 T^{3} - 254 T^{4} + 52 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 12 T^{2} + 40 T^{3} - 202 T^{4} + 40 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 68 T^{2} + 412 T^{3} + 2322 T^{4} + 412 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 136 T^{2} - 396 T^{3} + 7310 T^{4} - 396 p T^{5} + 136 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 80 T^{2} + 8 p T^{3} + 2722 T^{4} + 8 p^{2} T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 48 T^{2} - 404 T^{3} + 3518 T^{4} - 404 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 132 T^{2} - 980 T^{3} + 8402 T^{4} - 980 p T^{5} + 132 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 120 T^{2} - 364 T^{3} + 8750 T^{4} - 364 p T^{5} + 120 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 204 T^{2} - 1808 T^{3} + 16086 T^{4} - 1808 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 232 T^{2} - 2576 T^{3} + 20866 T^{4} - 2576 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 20 T + 304 T^{2} - 3460 T^{3} + 31774 T^{4} - 3460 p T^{5} + 304 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 24 T + 444 T^{2} + 5176 T^{3} + 51542 T^{4} + 5176 p T^{5} + 444 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 224 T^{2} - 1288 T^{3} + 21730 T^{4} - 1288 p T^{5} + 224 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 32 T + 612 T^{2} + 7840 T^{3} + 1002 p T^{4} + 7840 p T^{5} + 612 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 308 T^{2} + 1004 T^{3} + 37378 T^{4} + 1004 p T^{5} + 308 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 20 T + 168 T^{2} - 684 T^{3} + 46 p T^{4} - 684 p T^{5} + 168 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 184 T^{2} - 636 T^{3} + 24830 T^{4} - 636 p T^{5} + 184 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$