# Properties

 Label 4-128772-1.1-c1e2-0-0 Degree $4$ Conductor $128772$ Sign $1$ Analytic cond. $8.21061$ Root an. cond. $1.69275$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·3-s − 4-s + 9-s − 2·12-s + 10·13-s + 16-s + 8·19-s − 6·25-s − 4·27-s − 36-s + 20·39-s − 6·43-s + 2·48-s − 7·49-s − 10·52-s + 16·57-s + 4·61-s − 64-s + 73-s − 12·75-s − 8·76-s − 4·79-s − 11·81-s + 12·97-s + 6·100-s + 20·103-s + 4·108-s + ⋯
 L(s)  = 1 + 1.15·3-s − 1/2·4-s + 1/3·9-s − 0.577·12-s + 2.77·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s − 0.769·27-s − 1/6·36-s + 3.20·39-s − 0.914·43-s + 0.288·48-s − 49-s − 1.38·52-s + 2.11·57-s + 0.512·61-s − 1/8·64-s + 0.117·73-s − 1.38·75-s − 0.917·76-s − 0.450·79-s − 1.22·81-s + 1.21·97-s + 3/5·100-s + 1.97·103-s + 0.384·108-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$128772$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73$$ Sign: $1$ Analytic conductor: $$8.21061$$ Root analytic conductor: $$1.69275$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{128772} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 128772,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.378728304$$ $$L(\frac12)$$ $$\approx$$ $$2.378728304$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + T^{2}$$
3$C_2$ $$1 - 2 T + p T^{2}$$
7$C_2$ $$1 + p T^{2}$$
73$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 2 T + p T^{2} )$$
good5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
17$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
23$C_2^2$ $$1 - 42 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2^2$ $$1 + 86 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$