# Properties

 Label 4-112e2-1.1-c1e2-0-1 Degree $4$ Conductor $12544$ Sign $1$ Analytic cond. $0.799816$ Root an. cond. $0.945687$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 5-s − 9-s + 5·11-s + 4·13-s + 15-s − 17-s + 19-s + 3·23-s + 25-s + 7·31-s − 5·33-s + 9·37-s − 4·39-s − 4·41-s − 8·43-s + 45-s − 7·47-s − 7·49-s + 51-s + 5·53-s − 5·55-s − 57-s + 7·59-s + 3·61-s − 4·65-s + 67-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.447·5-s − 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s + 1.25·31-s − 0.870·33-s + 1.47·37-s − 0.640·39-s − 0.624·41-s − 1.21·43-s + 0.149·45-s − 1.02·47-s − 49-s + 0.140·51-s + 0.686·53-s − 0.674·55-s − 0.132·57-s + 0.911·59-s + 0.384·61-s − 0.496·65-s + 0.122·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$12544$$    =    $$2^{8} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$0.799816$$ Root analytic conductor: $$0.945687$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{12544} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 12544,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8721820425$$ $$L(\frac12)$$ $$\approx$$ $$0.8721820425$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2$ $$1 + p T^{2}$$
good3$D_{4}$ $$1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + T + p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$D_{4}$ $$1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 7 T + 38 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
41$C_4$ $$1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$D_{4}$ $$1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 7 T + 10 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$D_{4}$ $$1 - T - 54 T^{2} - p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
89$D_{4}$ $$1 + 13 T + 124 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$