# Properties

 Label 4-3319-1.1-c1e2-0-0 Degree $4$ Conductor $3319$ Sign $1$ Analytic cond. $0.211622$ Root an. cond. $0.678250$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $2$

# Origins

Among degree 4 primitive L-functions of analytic rank 2, this has the smallest conductor

## Dirichlet series

 L(s)  = 1 − 3·2-s − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 3·7-s − 3·8-s + 8·9-s + 15·10-s − 2·11-s − 16·12-s − 4·13-s + 9·14-s + 20·15-s + 3·16-s − 2·17-s − 24·18-s − 20·20-s + 12·21-s + 6·22-s − 23-s + 12·24-s + 12·25-s + 12·26-s − 12·27-s − 12·28-s − 8·29-s + ⋯
 L(s)  = 1 − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 1.13·7-s − 1.06·8-s + 8/3·9-s + 4.74·10-s − 0.603·11-s − 4.61·12-s − 1.10·13-s + 2.40·14-s + 5.16·15-s + 3/4·16-s − 0.485·17-s − 5.65·18-s − 4.47·20-s + 2.61·21-s + 1.27·22-s − 0.208·23-s + 2.44·24-s + 12/5·25-s + 2.35·26-s − 2.30·27-s − 2.26·28-s − 1.48·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3319$$ Sign: $1$ Analytic conductor: $$0.211622$$ Root analytic conductor: $$0.678250$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 3319,\ (\ :1/2, 1/2),\ 1)$$

## 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)$
bad3319$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 74 T + p T^{2} )$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$D_{4}$ $$1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
23$D_{4}$ $$1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 11 T + 65 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$