# Properties

 Label 4-7e2-1.1-c6e2-0-0 Degree $4$ Conductor $49$ Sign $1$ Analytic cond. $2.59331$ Root an. cond. $1.26900$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s + 64·4-s + 266·7-s + 1.02e3·8-s − 582·9-s + 1.74e3·11-s − 4.25e3·14-s − 1.63e4·16-s + 9.31e3·18-s − 2.79e4·22-s + 9.47e3·23-s + 2.92e4·25-s + 1.70e4·28-s + 2.22e4·29-s + 6.55e4·32-s − 3.72e4·36-s + 6.00e3·37-s + 6.28e4·43-s + 1.11e5·44-s − 1.51e5·46-s − 4.68e4·49-s − 4.67e5·50-s − 1.52e5·53-s + 2.72e5·56-s − 3.56e5·58-s − 1.54e5·63-s + 7.86e5·64-s + ⋯
 L(s)  = 1 − 2·2-s + 4-s + 0.775·7-s + 2·8-s − 0.798·9-s + 1.31·11-s − 1.55·14-s − 4·16-s + 1.59·18-s − 2.62·22-s + 0.778·23-s + 1.86·25-s + 0.775·28-s + 0.914·29-s + 2·32-s − 0.798·36-s + 0.118·37-s + 0.790·43-s + 1.31·44-s − 1.55·46-s − 0.398·49-s − 3.73·50-s − 1.02·53-s + 1.55·56-s − 1.82·58-s − 0.619·63-s + 3·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $1$ Analytic conductor: $$2.59331$$ Root analytic conductor: $$1.26900$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 49,\ (\ :3, 3),\ 1)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.4460537853$$ $$L(\frac12)$$ $$\approx$$ $$0.4460537853$$ $$L(4)$$ 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$C_2$ $$1 - 38 p T + p^{6} T^{2}$$
good2$C_2$ $$( 1 + p^{3} T + p^{6} T^{2} )^{2}$$
3$C_2^2$ $$1 + 194 p T^{2} + p^{12} T^{4}$$
5$C_2^2$ $$1 - 5842 p T^{2} + p^{12} T^{4}$$
11$C_2$ $$( 1 - 874 T + p^{6} T^{2} )^{2}$$
13$C_2^2$ $$1 - 4755578 T^{2} + p^{12} T^{4}$$
17$C_2^2$ $$1 - 748834 p T^{2} + p^{12} T^{4}$$
19$C_2^2$ $$1 - 84379322 T^{2} + p^{12} T^{4}$$
23$C_2$ $$( 1 - 206 p T + p^{6} T^{2} )^{2}$$
29$C_2$ $$( 1 - 11146 T + p^{6} T^{2} )^{2}$$
31$C_2^2$ $$1 - 1020892802 T^{2} + p^{12} T^{4}$$
37$C_2$ $$( 1 - 3002 T + p^{6} T^{2} )^{2}$$
41$C_2^2$ $$1 - 6189133442 T^{2} + p^{12} T^{4}$$
43$C_2$ $$( 1 - 31418 T + p^{6} T^{2} )^{2}$$
47$C_2^2$ $$1 - 16309886018 T^{2} + p^{12} T^{4}$$
53$C_2$ $$( 1 + 76406 T + p^{6} T^{2} )^{2}$$
59$C_2^2$ $$1 - 71539567322 T^{2} + p^{12} T^{4}$$
61$C_2^2$ $$1 - 27356175482 T^{2} + p^{12} T^{4}$$
67$C_2$ $$( 1 - 495242 T + p^{6} T^{2} )^{2}$$
71$C_2$ $$( 1 + 184406 T + p^{6} T^{2} )^{2}$$
73$C_2^2$ $$1 - 298950552578 T^{2} + p^{12} T^{4}$$
79$C_2$ $$( 1 + 534934 T + p^{6} T^{2} )^{2}$$
83$C_2^2$ $$1 - 142873131578 T^{2} + p^{12} T^{4}$$
89$C_2^2$ $$1 - 597656180162 T^{2} + p^{12} T^{4}$$
97$C_2^2$ $$1 - 1002631840898 T^{2} + p^{12} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$