# Properties

 Label 4-29e4-1.1-c1e2-0-2 Degree $4$ Conductor $707281$ Sign $1$ Analytic cond. $45.0968$ Root an. cond. $2.59141$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s − 2·4-s + 5-s − 6-s + 3·8-s − 4·9-s − 10-s − 5·11-s − 2·12-s + 4·13-s + 15-s + 16-s − 11·17-s + 4·18-s − 3·19-s − 2·20-s + 5·22-s + 2·23-s + 3·24-s + 2·25-s − 4·26-s − 6·27-s − 30-s − 9·31-s − 2·32-s − 5·33-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s − 4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s − 4/3·9-s − 0.316·10-s − 1.50·11-s − 0.577·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 2.66·17-s + 0.942·18-s − 0.688·19-s − 0.447·20-s + 1.06·22-s + 0.417·23-s + 0.612·24-s + 2/5·25-s − 0.784·26-s − 1.15·27-s − 0.182·30-s − 1.61·31-s − 0.353·32-s − 0.870·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$707281$$    =    $$29^{4}$$ Sign: $1$ Analytic conductor: $$45.0968$$ Root analytic conductor: $$2.59141$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 707281,\ (\ :1/2, 1/2),\ 1)$$

## 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)$
bad29 $$1$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 - T - T^{2} - p T^{3} + p^{2} T^{4}$$
7$C_2^2$ $$1 + 9 T^{2} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - T + 71 T^{2} - p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
47$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
59$D_{4}$ $$1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + T + 127 T^{2} + p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 4 T + 137 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$