# Properties

 Label 4-438869-1.1-c1e2-0-0 Degree $4$ Conductor $438869$ Sign $-1$ Analytic cond. $27.9826$ Root an. cond. $2.29997$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $3$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3·3-s − 4-s − 3·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 3·10-s − 6·11-s + 3·12-s − 3·13-s + 2·14-s + 9·15-s − 16-s − 2·17-s − 4·18-s − 8·19-s + 3·20-s + 6·21-s + 6·22-s − 2·23-s − 3·24-s + 3·26-s − 6·27-s + 2·28-s − 8·29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 0.948·10-s − 1.80·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 2.32·15-s − 1/4·16-s − 0.485·17-s − 0.942·18-s − 1.83·19-s + 0.670·20-s + 1.30·21-s + 1.27·22-s − 0.417·23-s − 0.612·24-s + 0.588·26-s − 1.15·27-s + 0.377·28-s − 1.48·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$438869$$ Sign: $-1$ Analytic conductor: $$27.9826$$ Root analytic conductor: $$2.29997$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{438869} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(4,\ 438869,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad438869$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 515 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_4$ $$1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$D_{4}$ $$1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$D_{4}$ $$1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
37$D_{4}$ $$1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 28 T^{2} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 13 T + 118 T^{2} - 13 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + T - 94 T^{2} + p T^{3} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$D_{4}$ $$1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
89$D_{4}$ $$1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 15 T + 130 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$