# Properties

 Label 4-7889-1.1-c1e2-0-0 Degree $4$ Conductor $7889$ Sign $1$ Analytic cond. $0.503009$ Root an. cond. $0.842158$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·4-s + 7-s + 4·9-s + 3·11-s + 12·16-s + 2·23-s − 25-s − 4·28-s − 3·29-s − 16·36-s − 5·37-s + 7·43-s − 12·44-s + 49-s − 3·53-s + 4·63-s − 32·64-s − 17·67-s + 3·77-s − 2·79-s + 7·81-s − 8·92-s + 12·99-s + 4·100-s − 15·107-s + 4·109-s + 12·112-s + ⋯
 L(s)  = 1 − 2·4-s + 0.377·7-s + 4/3·9-s + 0.904·11-s + 3·16-s + 0.417·23-s − 1/5·25-s − 0.755·28-s − 0.557·29-s − 8/3·36-s − 0.821·37-s + 1.06·43-s − 1.80·44-s + 1/7·49-s − 0.412·53-s + 0.503·63-s − 4·64-s − 2.07·67-s + 0.341·77-s − 0.225·79-s + 7/9·81-s − 0.834·92-s + 1.20·99-s + 2/5·100-s − 1.45·107-s + 0.383·109-s + 1.13·112-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$7889$$    =    $$7^{3} \cdot 23$$ Sign: $1$ Analytic conductor: $$0.503009$$ Root analytic conductor: $$0.842158$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{7889} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 7889,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7285964346$$ $$L(\frac12)$$ $$\approx$$ $$0.7285964346$$ $$L(\frac{3}{2})$$ 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)$
bad7$C_1$ $$1 - T$$
23$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 3 T + p T^{2} )$$
good2$C_2$ $$( 1 + p T^{2} )^{2}$$
3$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 29 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 7 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2^2$ $$1 + 28 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
41$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
47$C_2^2$ $$1 - 68 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2^2$ $$1 - 44 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
71$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
73$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2^2$ $$1 + 49 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 115 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 + 13 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$