# Properties

 Label 4-882e2-1.1-c3e2-0-36 Degree $4$ Conductor $777924$ Sign $1$ Analytic cond. $2708.12$ Root an. cond. $7.21385$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·5-s − 8·8-s + 4·10-s − 8·11-s − 84·13-s − 16·16-s − 2·17-s + 124·19-s − 16·22-s + 76·23-s + 125·25-s − 168·26-s − 508·29-s + 72·31-s − 4·34-s − 398·37-s + 248·38-s − 16·40-s − 924·41-s + 424·43-s + 152·46-s − 264·47-s + 250·50-s − 162·53-s − 16·55-s − 1.01e3·58-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.178·5-s − 0.353·8-s + 0.126·10-s − 0.219·11-s − 1.79·13-s − 1/4·16-s − 0.0285·17-s + 1.49·19-s − 0.155·22-s + 0.689·23-s + 25-s − 1.26·26-s − 3.25·29-s + 0.417·31-s − 0.0201·34-s − 1.76·37-s + 1.05·38-s − 0.0632·40-s − 3.51·41-s + 1.50·43-s + 0.487·46-s − 0.819·47-s + 0.707·50-s − 0.419·53-s − 0.0392·55-s − 2.30·58-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$777924$$    =    $$2^{2} \cdot 3^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$2708.12$$ Root analytic conductor: $$7.21385$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 777924,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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$C_2$ $$1 - p T + p^{2} T^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^2$ $$1 - 2 T - 121 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 8 T - 1267 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}$$
13$C_2$ $$( 1 + 42 T + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 + 2 T - 4909 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 76 T - 6391 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2$ $$( 1 + 254 T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 - 72 T - 24607 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4}$$
37$C_2^2$ $$1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2$ $$( 1 + 462 T + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 - 212 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 + 264 T - 34127 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4}$$
53$C_2^2$ $$1 + 162 T - 122633 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2^2$ $$1 + 772 T + 390605 T^{2} + 772 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2^2$ $$1 + 30 T - 226081 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4}$$
67$C_2^2$ $$1 - 764 T + 282933 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 - 236 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 418 T - 214293 T^{2} + 418 p^{3} T^{3} + p^{6} T^{4}$$
79$C_2^2$ $$1 + 552 T - 188335 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4}$$
83$C_2$ $$( 1 + 1036 T + p^{3} T^{2} )^{2}$$
89$C_2^2$ $$1 - 30 T - 704069 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2$ $$( 1 + 1190 T + p^{3} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$