# Properties

 Label 4-7e4-1.1-c3e2-0-2 Degree $4$ Conductor $2401$ Sign $1$ Analytic cond. $8.35842$ Root an. cond. $1.70032$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5·2-s + 8·4-s − 5·8-s + 27·9-s + 68·11-s − 25·16-s + 135·18-s + 340·22-s + 40·23-s + 125·25-s − 332·29-s − 40·32-s + 216·36-s − 450·37-s − 360·43-s + 544·44-s + 200·46-s + 625·50-s − 590·53-s − 1.66e3·58-s − 487·64-s + 740·67-s + 1.37e3·71-s − 135·72-s − 2.25e3·74-s + 1.38e3·79-s − 1.80e3·86-s + ⋯
 L(s)  = 1 + 1.76·2-s + 4-s − 0.220·8-s + 9-s + 1.86·11-s − 0.390·16-s + 1.76·18-s + 3.29·22-s + 0.362·23-s + 25-s − 2.12·29-s − 0.220·32-s + 36-s − 1.99·37-s − 1.27·43-s + 1.86·44-s + 0.641·46-s + 1.76·50-s − 1.52·53-s − 3.75·58-s − 0.951·64-s + 1.34·67-s + 2.30·71-s − 0.220·72-s − 3.53·74-s + 1.97·79-s − 2.25·86-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2401$$    =    $$7^{4}$$ Sign: $1$ Analytic conductor: $$8.35842$$ Root analytic conductor: $$1.70032$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{49} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2401,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.968010391$$ $$L(\frac12)$$ $$\approx$$ $$3.968010391$$ $$L(\frac{5}{2})$$ 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 $$1$$
good2$C_2^2$ $$1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}$$
3$C_2$ $$( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} )$$
5$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
11$C_2^2$ $$1 - 68 T + 3293 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4}$$
13$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
19$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 40 T - 10567 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2$ $$( 1 + 166 T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
37$C_2^2$ $$1 + 450 T + 151847 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 + 180 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 + 590 T + 199223 T^{2} + 590 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
61$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
67$C_2^2$ $$1 - 740 T + 246837 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 - 688 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
79$C_2^2$ $$1 - 1384 T + 1422417 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4}$$
83$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
89$C_2^2$ $$1 - p^{3} T^{2} + p^{6} T^{4}$$
97$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$