# Properties

 Label 8-12e12-1.1-c2e4-0-13 Degree $8$ Conductor $8.916\times 10^{12}$ Sign $1$ Analytic cond. $4.91490\times 10^{6}$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 18·5-s + 2·7-s + 18·11-s + 10·13-s + 40·19-s − 18·23-s + 139·25-s + 18·29-s + 38·31-s − 36·35-s − 128·37-s + 126·41-s + 46·43-s − 54·47-s + 45·49-s − 324·55-s + 126·59-s − 62·61-s − 180·65-s + 106·67-s − 208·73-s + 36·77-s + 14·79-s − 378·83-s + 20·91-s − 720·95-s + 14·97-s + ⋯
 L(s)  = 1 − 3.59·5-s + 2/7·7-s + 1.63·11-s + 0.769·13-s + 2.10·19-s − 0.782·23-s + 5.55·25-s + 0.620·29-s + 1.22·31-s − 1.02·35-s − 3.45·37-s + 3.07·41-s + 1.06·43-s − 1.14·47-s + 0.918·49-s − 5.89·55-s + 2.13·59-s − 1.01·61-s − 2.76·65-s + 1.58·67-s − 2.84·73-s + 0.467·77-s + 0.177·79-s − 4.55·83-s + 0.219·91-s − 7.57·95-s + 0.144·97-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 3^{12}$$ Sign: $1$ Analytic conductor: $$4.91490\times 10^{6}$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.901142653$$ $$L(\frac12)$$ $$\approx$$ $$1.901142653$$ $$L(2)$$ 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 $$1$$
good5$C_2^2$ $$( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
7$D_4\times C_2$ $$1 - 2 T - 41 T^{2} + 106 T^{3} - 572 T^{4} + 106 p^{2} T^{5} - 41 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8}$$
11$D_4\times C_2$ $$1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 4518 p^{2} T^{5} + 359 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8}$$
13$D_4\times C_2$ $$1 - 10 T - 47 T^{2} + 1910 T^{3} - 23852 T^{4} + 1910 p^{2} T^{5} - 47 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8}$$
17$D_4\times C_2$ $$1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8}$$
19$D_{4}$ $$( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 + 18 T + 1175 T^{2} + 19206 T^{3} + 915780 T^{4} + 19206 p^{2} T^{5} + 1175 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8}$$
29$D_4\times C_2$ $$1 - 18 T + 1745 T^{2} - 29466 T^{3} + 2063316 T^{4} - 29466 p^{2} T^{5} + 1745 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8}$$
31$D_4\times C_2$ $$1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4750 p^{2} T^{5} - 353 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 + 64 T + 3546 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 126 T + 9329 T^{2} - 508662 T^{3} + 22367460 T^{4} - 508662 p^{2} T^{5} + 9329 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8}$$
43$D_4\times C_2$ $$1 - 46 T - 1625 T^{2} - 46 p T^{3} + 3604 p^{2} T^{4} - 46 p^{3} T^{5} - 1625 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8}$$
47$D_4\times C_2$ $$1 + 54 T + 4751 T^{2} + 204066 T^{3} + 11548308 T^{4} + 204066 p^{2} T^{5} + 4751 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 126 T + 10535 T^{2} - 660618 T^{3} + 33793140 T^{4} - 660618 p^{2} T^{5} + 10535 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8}$$
61$D_4\times C_2$ $$1 + 62 T - 2615 T^{2} - 60946 T^{3} + 13569316 T^{4} - 60946 p^{2} T^{5} - 2615 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8}$$
67$D_4\times C_2$ $$1 - 106 T - 65 T^{2} - 246238 T^{3} + 57123076 T^{4} - 246238 p^{2} T^{5} - 65 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8}$$
71$D_4\times C_2$ $$1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 14 T - 10985 T^{2} + 18214 T^{3} + 84841444 T^{4} + 18214 p^{2} T^{5} - 10985 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8}$$
83$D_4\times C_2$ $$1 + 378 T + 72863 T^{2} + 9538830 T^{3} + 917456196 T^{4} + 9538830 p^{2} T^{5} + 72863 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8}$$
97$D_4\times C_2$ $$1 - 14 T - 8087 T^{2} + 147490 T^{3} - 21765356 T^{4} + 147490 p^{2} T^{5} - 8087 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$