# Properties

 Label 4-180e2-1.1-c1e2-0-7 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 − 2-s − 4-s + 7-s + 3·8-s − 3·9-s − 5·11-s + 13-s − 14-s − 16-s + 5·17-s + 3·18-s − 4·19-s + 5·22-s − 3·23-s − 5·25-s − 26-s − 28-s − 4·29-s − 5·31-s − 5·32-s − 5·34-s + 3·36-s + 9·37-s + 4·38-s − 7·41-s + 3·43-s + 5·44-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 9-s − 1.50·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.21·17-s + 0.707·18-s − 0.917·19-s + 1.06·22-s − 0.625·23-s − 25-s − 0.196·26-s − 0.188·28-s − 0.742·29-s − 0.898·31-s − 0.883·32-s − 0.857·34-s + 1/2·36-s + 1.47·37-s + 0.648·38-s − 1.09·41-s + 0.457·43-s + 0.753·44-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$C_2$ $$1 + T + p T^{2}$$
3$C_2$ $$1 + p T^{2}$$
5$C_2$ $$1 + p T^{2}$$
good7$C_4$ $$1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$D_{4}$ $$1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$D_{4}$ $$1 - 9 T + 91 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 7 T + 61 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 3 T + T^{2} - 3 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 5 T - 2 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 19 T + 214 T^{2} + 19 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$