# Properties

 Label 8-7605e4-1.1-c1e4-0-2 Degree $8$ Conductor $3.345\times 10^{15}$ Sign $1$ Analytic cond. $1.35989\times 10^{7}$ Root an. cond. $7.79270$ 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-s + 4·5-s + 10·7-s + 4·8-s − 8·10-s − 20·14-s + 2·17-s + 16·19-s − 4·20-s + 10·23-s + 10·25-s − 10·28-s − 8·29-s + 8·31-s − 2·32-s − 4·34-s + 40·35-s − 2·37-s − 32·38-s + 16·40-s − 8·41-s − 2·43-s − 20·46-s − 8·47-s + 42·49-s − 20·50-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1/2·4-s + 1.78·5-s + 3.77·7-s + 1.41·8-s − 2.52·10-s − 5.34·14-s + 0.485·17-s + 3.67·19-s − 0.894·20-s + 2.08·23-s + 2·25-s − 1.88·28-s − 1.48·29-s + 1.43·31-s − 0.353·32-s − 0.685·34-s + 6.76·35-s − 0.328·37-s − 5.19·38-s + 2.52·40-s − 1.24·41-s − 0.304·43-s − 2.94·46-s − 1.16·47-s + 6·49-s − 2.82·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{4} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$1.35989\times 10^{7}$$ Root analytic conductor: $$7.79270$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{7605} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$13.32981862$$ $$L(\frac12)$$ $$\approx$$ $$13.32981862$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 - T )^{4}$$
13 $$1$$
good2$D_4\times C_2$ $$1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
7$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 58 T^{2} - 232 T^{3} + 703 T^{4} - 232 p T^{5} + 58 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2^2 \wr C_2$ $$1 + 14 T^{2} + 9 p T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 2 T + 50 T^{2} - 92 T^{3} + 1135 T^{4} - 92 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 98 T^{2} - 544 T^{3} + 137 p T^{4} - 544 p T^{5} + 98 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 98 T^{2} + 656 T^{3} + 4003 T^{4} + 656 p T^{5} + 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
31$D_{4}$ $$( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 94 T^{2} + 260 T^{3} + 4219 T^{4} + 260 p T^{5} + 94 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 4 T + 83 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 154 T^{2} + 248 T^{3} + 9559 T^{4} + 248 p T^{5} + 154 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 116 T^{2} + 392 T^{3} + 5158 T^{4} + 392 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 248 T^{2} - 36 p T^{3} + 20622 T^{4} - 36 p^{2} T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 12 T + 266 T^{2} + 2136 T^{3} + 24423 T^{4} + 2136 p T^{5} + 266 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 28 T + 502 T^{2} - 6088 T^{3} + 55063 T^{4} - 6088 p T^{5} + 502 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 30 T + 598 T^{2} - 7608 T^{3} + 73923 T^{4} - 7608 p T^{5} + 598 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 74 T^{2} + 424 T^{3} + 10903 T^{4} + 424 p T^{5} + 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 208 T^{2} + 920 T^{3} + 17998 T^{4} + 920 p T^{5} + 208 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 308 T^{2} - 2700 T^{3} + 37158 T^{4} - 2700 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 122 T^{2} - 1056 T^{3} + 14727 T^{4} - 1056 p T^{5} + 122 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 298 T^{2} - 956 T^{3} + 38551 T^{4} - 956 p T^{5} + 298 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$