# Properties

 Label 4-180e2-1.1-c1e2-0-8 Degree $4$ Conductor $32400$ Sign $-1$ Analytic cond. $2.06585$ Root an. cond. $1.19887$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 5·7-s + 9-s + 2·11-s + 13-s − 2·17-s − 19-s + 5·21-s − 10·23-s + 25-s − 27-s − 4·31-s − 2·33-s − 37-s − 39-s − 8·41-s − 4·43-s + 11·49-s + 2·51-s − 4·53-s + 57-s + 8·59-s + 5·61-s − 5·63-s + 5·67-s + 10·69-s − 22·71-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 0.229·19-s + 1.09·21-s − 2.08·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s − 0.348·33-s − 0.164·37-s − 0.160·39-s − 1.24·41-s − 0.609·43-s + 11/7·49-s + 0.280·51-s − 0.549·53-s + 0.132·57-s + 1.04·59-s + 0.640·61-s − 0.629·63-s + 0.610·67-s + 1.20·69-s − 2.61·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$32400$$    =    $$2^{4} \cdot 3^{4} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$2.06585$$ Root analytic conductor: $$1.19887$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 32400,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_1$ $$1 + T$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$D_{4}$ $$1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$D_{4}$ $$1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 5 T + 106 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 7 T + 24 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$