# Properties

 Label 40-11e60-1.1-c0e20-0-0 Degree $40$ Conductor $3.045\times 10^{62}$ Sign $1$ Analytic cond. $0.000279707$ Root an. cond. $0.815018$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯
 L(s)  = 1 + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$40$$ Conductor: $$11^{60}$$ Sign: $1$ Analytic conductor: $$0.000279707$$ Root analytic conductor: $$0.815018$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1331} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(40,\ 11^{60} ,\ ( \ : [0]^{20} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.01410600819$$ $$L(\frac12)$$ $$\approx$$ $$0.01410600819$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1$$
good2 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
3 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
5 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
7 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
13 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
17 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
19 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
23 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4}$$
29 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
31 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
37 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
41 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
43 $$( 1 - T )^{20}( 1 + T )^{20}$$
47 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
53 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
59 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
61 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
67 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4}$$
71 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
73 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
79 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
83 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
89 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4}$$
97 $$1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$