# Properties

 Label 8-74e4-1.1-c1e4-0-0 Degree $8$ Conductor $29986576$ Sign $1$ Analytic cond. $0.121908$ Root an. cond. $0.768695$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 4-s − 6·5-s − 8·7-s + 4·9-s + 12·11-s − 2·12-s + 12·15-s − 6·17-s + 6·19-s − 6·20-s + 16·21-s + 11·25-s − 4·27-s − 8·28-s − 24·33-s + 48·35-s + 4·36-s − 2·37-s + 6·41-s + 12·44-s − 24·45-s + 12·47-s + 30·49-s + 12·51-s + 12·53-s − 72·55-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1/2·4-s − 2.68·5-s − 3.02·7-s + 4/3·9-s + 3.61·11-s − 0.577·12-s + 3.09·15-s − 1.45·17-s + 1.37·19-s − 1.34·20-s + 3.49·21-s + 11/5·25-s − 0.769·27-s − 1.51·28-s − 4.17·33-s + 8.11·35-s + 2/3·36-s − 0.328·37-s + 0.937·41-s + 1.80·44-s − 3.57·45-s + 1.75·47-s + 30/7·49-s + 1.68·51-s + 1.64·53-s − 9.70·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$29986576$$    =    $$2^{4} \cdot 37^{4}$$ Sign: $1$ Analytic conductor: $$0.121908$$ Root analytic conductor: $$0.768695$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 29986576,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2908499195$$ $$L(\frac12)$$ $$\approx$$ $$0.2908499195$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
37$C_2$ $$( 1 + T + p T^{2} )^{2}$$
good3$D_4\times C_2$ $$1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8}$$
5$C_2^2$ $$( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2$ $$( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2}$$
11$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$$\times$$C_2^2$ $$( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )$$
17$C_2^2$ $$( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 6 T - 43 T^{2} + 18 T^{3} + 3084 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
47$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 12 T + 14 T^{2} - 288 T^{3} + 6459 T^{4} - 288 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 12 T + 142 T^{2} - 1128 T^{3} + 8187 T^{4} - 1128 p T^{5} + 142 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2$$\times$$C_2^2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )$$
67$D_4\times C_2$ $$1 + 2 T - 104 T^{2} - 52 T^{3} + 6907 T^{4} - 52 p T^{5} - 104 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 + 12 T - 22 T^{2} + 288 T^{3} + 12291 T^{4} + 288 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
73$D_{4}$ $$( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 + 18 T + 284 T^{2} + 3168 T^{3} + 33267 T^{4} + 3168 p T^{5} + 284 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 + 6 T - 64 T^{2} - 396 T^{3} + 123 T^{4} - 396 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 9 T + 116 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$