# Properties

 Label 4-1142784-1.1-c1e2-0-8 Degree $4$ Conductor $1142784$ Sign $-1$ Analytic cond. $72.8648$ Root an. cond. $2.92165$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 7-s − 2·9-s + 3·13-s − 19-s + 21-s + 25-s − 5·27-s − 4·31-s − 9·37-s + 3·39-s + 2·43-s − 11·49-s − 57-s + 9·61-s − 2·63-s − 6·67-s − 11·73-s + 75-s + 79-s + 81-s + 3·91-s − 4·93-s − 6·97-s + 21·103-s + 7·109-s − 9·111-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.832·13-s − 0.229·19-s + 0.218·21-s + 1/5·25-s − 0.962·27-s − 0.718·31-s − 1.47·37-s + 0.480·39-s + 0.304·43-s − 1.57·49-s − 0.132·57-s + 1.15·61-s − 0.251·63-s − 0.733·67-s − 1.28·73-s + 0.115·75-s + 0.112·79-s + 1/9·81-s + 0.314·91-s − 0.414·93-s − 0.609·97-s + 2.06·103-s + 0.670·109-s − 0.854·111-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1142784$$    =    $$2^{12} \cdot 3^{2} \cdot 31$$ Sign: $-1$ Analytic conductor: $$72.8648$$ Root analytic conductor: $$2.92165$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1142784} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 1142784,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_2$ $$1 - T + p T^{2}$$
31$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 5 T + p T^{2} )$$
good5$C_2^2$ $$1 - T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
17$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$C_2^2$ $$1 - 25 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2^2$ $$1 + 9 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
47$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 74 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 58 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
71$C_2^2$ $$1 + 100 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
83$C_2^2$ $$1 + 35 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 12 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$