# Properties

 Label 4-112e2-1.1-c1e2-0-8 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 $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·3-s + 8·9-s − 2·17-s − 4·19-s − 8·25-s − 12·27-s − 10·41-s − 7·49-s + 8·51-s + 16·57-s − 20·59-s − 10·73-s + 32·75-s + 23·81-s + 20·83-s + 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·121-s + 40·123-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 2.30·3-s + 8/3·9-s − 0.485·17-s − 0.917·19-s − 8/5·25-s − 2.30·27-s − 1.56·41-s − 49-s + 1.12·51-s + 2.11·57-s − 2.60·59-s − 1.17·73-s + 3.69·75-s + 23/9·81-s + 2.19·83-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 6/11·121-s + 3.60·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: $$1$$ Selberg data: $$(4,\ 12544,\ (\ :1/2, 1/2),\ -1)$$

## 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 $$1$$
7$C_2$ $$1 + p T^{2}$$
good3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
23$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2^2$ $$1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 70 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 62 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2^2$ $$1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 72 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$