# Properties

 Label 8-1875e4-1.1-c1e4-0-2 Degree $8$ Conductor $1.236\times 10^{13}$ Sign $1$ Analytic cond. $50247.3$ Root an. cond. $3.86936$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4·3-s + 2·4-s + 8·6-s + 2·7-s + 5·8-s + 10·9-s − 7·11-s + 8·12-s + 13-s + 4·14-s + 5·16-s + 2·17-s + 20·18-s + 5·19-s + 8·21-s − 14·22-s + 23-s + 20·24-s + 2·26-s + 20·27-s + 4·28-s − 20·29-s + 23·31-s − 2·32-s − 28·33-s + 4·34-s + ⋯
 L(s)  = 1 + 1.41·2-s + 2.30·3-s + 4-s + 3.26·6-s + 0.755·7-s + 1.76·8-s + 10/3·9-s − 2.11·11-s + 2.30·12-s + 0.277·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s + 4.71·18-s + 1.14·19-s + 1.74·21-s − 2.98·22-s + 0.208·23-s + 4.08·24-s + 0.392·26-s + 3.84·27-s + 0.755·28-s − 3.71·29-s + 4.13·31-s − 0.353·32-s − 4.87·33-s + 0.685·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$50247.3$$ Root analytic conductor: $$3.86936$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$35.63305874$$ $$L(\frac12)$$ $$\approx$$ $$35.63305874$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_1$ $$( 1 - T )^{4}$$
5 $$1$$
good2$C_2^2:C_4$ $$1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
7$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 12 T^{2} - 15 T^{3} + 61 T^{4} - 15 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 7 T + 38 T^{2} + 139 T^{3} + 485 T^{4} + 139 p T^{5} + 38 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - T + 38 T^{2} - 15 T^{3} + 641 T^{4} - 15 p T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 62 T^{2} - 95 T^{3} + 1541 T^{4} - 95 p T^{5} + 62 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 5 T + 41 T^{2} - 210 T^{3} + 1111 T^{4} - 210 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - T + 68 T^{2} - 5 p T^{3} + 2051 T^{4} - 5 p^{2} T^{5} + 68 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 + 20 T + 256 T^{2} + 2145 T^{3} + 13571 T^{4} + 2145 p T^{5} + 256 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 23 T + 308 T^{2} - 2751 T^{3} + 17885 T^{4} - 2751 p T^{5} + 308 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 102 T^{2} - 175 T^{3} + 4811 T^{4} - 175 p T^{5} + 102 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 233 T^{2} - 2040 T^{3} + 16241 T^{4} - 2040 p T^{5} + 233 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 152 T^{2} - 245 T^{3} + 10181 T^{4} - 245 p T^{5} + 152 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 178 T^{2} + 585 T^{3} + 13511 T^{4} + 585 p T^{5} + 178 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 15 T + 191 T^{2} + 1440 T^{3} + 11931 T^{4} + 1440 p T^{5} + 191 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 188 T^{2} + 444 T^{3} + 15485 T^{4} + 444 p T^{5} + 188 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 237 T^{2} - 390 T^{3} + 22951 T^{4} - 390 p T^{5} + 237 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 68 T^{2} - 811 T^{3} + 485 T^{4} - 811 p T^{5} + 68 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 243 T^{2} - 2010 T^{3} + 19951 T^{4} - 2010 p T^{5} + 243 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 35 T + 636 T^{2} - 7945 T^{3} + 78161 T^{4} - 7945 p T^{5} + 636 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 278 T^{2} - 2680 T^{3} + 31391 T^{4} - 2680 p T^{5} + 278 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 35 T + 776 T^{2} + 11375 T^{3} + 125591 T^{4} + 11375 p T^{5} + 776 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 302 T^{2} - 2760 T^{3} + 41871 T^{4} - 2760 p T^{5} + 302 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$