# Properties

 Label 4-771228-1.1-c1e2-0-2 Degree $4$ Conductor $771228$ Sign $1$ Analytic cond. $49.1741$ Root an. cond. $2.64809$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $2$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 4-s − 2·7-s + 9-s + 12-s − 6·13-s + 16-s − 8·19-s + 2·21-s − 4·25-s − 27-s + 2·28-s − 14·31-s − 36-s + 3·37-s + 6·39-s − 10·43-s − 48-s − 10·49-s + 6·52-s + 8·57-s + 4·61-s − 2·63-s − 64-s − 12·67-s − 10·73-s + 4·75-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s − 4/5·25-s − 0.192·27-s + 0.377·28-s − 2.51·31-s − 1/6·36-s + 0.493·37-s + 0.960·39-s − 1.52·43-s − 0.144·48-s − 1.42·49-s + 0.832·52-s + 1.05·57-s + 0.512·61-s − 0.251·63-s − 1/8·64-s − 1.46·67-s − 1.17·73-s + 0.461·75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 771228 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 771228 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$771228$$    =    $$2^{2} \cdot 3^{3} \cdot 37 \cdot 193$$ Sign: $1$ Analytic conductor: $$49.1741$$ Root analytic conductor: $$2.64809$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{771228} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 771228,\ (\ :1/2, 1/2),\ 1)$$

## 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$C_2$ $$1 + T^{2}$$
3$C_1$ $$1 + T$$
37$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 2 T + p T^{2} )$$
193$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
good5$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2^2$ $$1 + 16 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 52 T^{2} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2^2$ $$1 - 80 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2^2$ $$1 + 54 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 100 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
71$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 + 106 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 152 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$