# Properties

 Degree $4$ Conductor $2160900$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 5-s − 6-s − 8-s + 10-s + 4·13-s − 15-s − 16-s − 6·17-s + 4·19-s + 24-s + 4·26-s + 27-s − 12·29-s − 30-s − 8·31-s − 6·34-s − 2·37-s + 4·38-s − 4·39-s − 40-s − 12·41-s − 8·43-s + 48-s + 6·51-s + 6·53-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 2.22·29-s − 0.182·30-s − 1.43·31-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.640·39-s − 0.158·40-s − 1.87·41-s − 1.21·43-s + 0.144·48-s + 0.840·51-s + 0.824·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2160900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1470} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2160900,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.605079590$$ $$L(\frac12)$$ $$\approx$$ $$1.605079590$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 - T + T^{2}$$
3$C_2$ $$1 + T + T^{2}$$
5$C_2$ $$1 - T + T^{2}$$
7 $$1$$
good11$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
23$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2^2$ $$1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2^2$ $$1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$