# Properties

 Label 8-1950e4-1.1-c1e4-0-31 Degree $8$ Conductor $1.446\times 10^{13}$ Sign $1$ Analytic cond. $58782.3$ Root an. cond. $3.94598$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 9-s − 2·11-s − 10·19-s + 40·31-s + 36-s − 12·41-s − 2·44-s − 5·49-s + 8·59-s + 4·61-s − 64-s + 4·71-s − 10·76-s + 40·79-s + 2·89-s − 2·99-s − 8·101-s + 40·109-s + 23·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 1/2·4-s + 1/3·9-s − 0.603·11-s − 2.29·19-s + 7.18·31-s + 1/6·36-s − 1.87·41-s − 0.301·44-s − 5/7·49-s + 1.04·59-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 1.14·76-s + 4.50·79-s + 0.211·89-s − 0.201·99-s − 0.796·101-s + 3.83·109-s + 2.09·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-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^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$58782.3$$ Root analytic conductor: $$3.94598$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1950} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.888630841$$ $$L(\frac12)$$ $$\approx$$ $$4.888630841$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
3$C_2^2$ $$1 - T^{2} + T^{4}$$
5 $$1$$
13$C_2^2$ $$1 - T^{2} + p^{2} T^{4}$$
good7$C_2^3$ $$1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 10 T + p T^{2} )^{4}$$
37$C_2^2$$\times$$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} )$$
41$C_2^2$ $$( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^3$ $$1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 - 13 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 63 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 10 T + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^3$ $$1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$