# Properties

 Label 8-2736e4-1.1-c1e4-0-5 Degree $8$ Conductor $5.604\times 10^{13}$ Sign $1$ Analytic cond. $227810.$ Root an. cond. $4.67408$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·7-s + 6·13-s + 12·17-s − 16·19-s − 8·25-s − 2·43-s + 12·47-s − 6·49-s − 24·59-s + 2·61-s + 6·67-s − 12·71-s + 14·73-s − 18·79-s + 24·91-s − 36·97-s − 36·101-s − 24·107-s + 48·119-s + 4·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 1.51·7-s + 1.66·13-s + 2.91·17-s − 3.67·19-s − 8/5·25-s − 0.304·43-s + 1.75·47-s − 6/7·49-s − 3.12·59-s + 0.256·61-s + 0.733·67-s − 1.42·71-s + 1.63·73-s − 2.02·79-s + 2.51·91-s − 3.65·97-s − 3.58·101-s − 2.32·107-s + 4.40·119-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{8} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$227810.$$ Root analytic conductor: $$4.67408$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2736} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.221399333$$ $$L(\frac12)$$ $$\approx$$ $$1.221399333$$ $$L(\frac{3}{2})$$ 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 $$1$$
3 $$1$$
19$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
good5$C_2^3$ $$1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8}$$
7$D_{4}$ $$( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 12 T + 92 T^{2} - 528 T^{3} + 2463 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2^3$ $$1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^3$ $$1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 + 2 T - 59 T^{2} - 46 T^{3} + 1948 T^{4} - 46 p T^{5} - 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 12 T + 152 T^{2} - 1248 T^{3} + 10863 T^{4} - 1248 p T^{5} + 152 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^3$ $$1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 24 T + 320 T^{2} + 3312 T^{3} + 28071 T^{4} + 3312 p T^{5} + 320 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 6 T + 77 T^{2} - 390 T^{3} + 540 T^{4} - 390 p T^{5} + 77 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 18 T + 275 T^{2} + 3006 T^{3} + 30180 T^{4} + 3006 p T^{5} + 275 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2^3$ $$1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$