# Properties

 Label 4-930e2-1.1-c1e2-0-13 Degree $4$ Conductor $864900$ Sign $1$ Analytic cond. $55.1467$ Root an. cond. $2.72508$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s − 4·7-s + 4·8-s + 2·10-s + 3·11-s + 3·12-s + 2·13-s − 8·14-s + 15-s + 5·16-s − 3·17-s + 3·20-s − 4·21-s + 6·22-s + 6·23-s + 4·24-s + 4·26-s − 27-s − 12·28-s + 4·29-s + 2·30-s + 4·31-s + 6·32-s + ⋯
 L(s)  = 1 + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s + 0.632·10-s + 0.904·11-s + 0.866·12-s + 0.554·13-s − 2.13·14-s + 0.258·15-s + 5/4·16-s − 0.727·17-s + 0.670·20-s − 0.872·21-s + 1.27·22-s + 1.25·23-s + 0.816·24-s + 0.784·26-s − 0.192·27-s − 2.26·28-s + 0.742·29-s + 0.365·30-s + 0.718·31-s + 1.06·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$864900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$$ Sign: $1$ Analytic conductor: $$55.1467$$ Root analytic conductor: $$2.72508$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 864900,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.555500076$$ $$L(\frac12)$$ $$\approx$$ $$6.555500076$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{2}$$
3$C_2$ $$1 - T + T^{2}$$
5$C_2$ $$1 - T + T^{2}$$
31$C_2$ $$1 - 4 T + p T^{2}$$
good7$C_2$ $$( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$C_2^2$ $$1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
17$C_2^2$ $$1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
41$C_2^2$ $$1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4}$$
47$C_2$ $$( 1 + T + p T^{2} )^{2}$$
53$C_2^2$ $$1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
71$C_2^2$ $$1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$C_2^2$ $$1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
79$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2^2$ $$1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
89$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 14 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$