# Properties

 Label 4-349-1.1-c1e2-0-0 Degree $4$ Conductor $349$ Sign $1$ Analytic cond. $0.0222525$ Root an. cond. $0.386229$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 4·8-s + 2·9-s + 2·10-s + 11-s − 2·12-s + 2·13-s + 4·14-s + 15-s + 8·16-s + 3·17-s − 4·18-s − 2·20-s + 2·21-s − 2·22-s − 3·23-s + 4·24-s − 6·25-s − 4·26-s − 6·27-s − 4·28-s + 29-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 2·16-s + 0.727·17-s − 0.942·18-s − 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.625·23-s + 0.816·24-s − 6/5·25-s − 0.784·26-s − 1.15·27-s − 0.755·28-s + 0.185·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$349$$ Sign: $1$ Analytic conductor: $$0.0222525$$ Root analytic conductor: $$0.386229$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 349,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1656123320$$ $$L(\frac12)$$ $$\approx$$ $$0.1656123320$$ $$L(\frac{3}{2})$$ 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)$
bad349$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 12 T + p T^{2} )$$
good2$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + T - T^{2} + p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$D_{4}$ $$1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$D_{4}$ $$1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 9 T + 42 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 11 T + p T^{2} - 11 p T^{3} + p^{2} T^{4}$$
71$C_2^2$ $$1 + 80 T^{2} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 11 T + 112 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + T + 86 T^{2} + p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$