# Properties

 Label 4-38e2-1.1-c5e2-0-0 Degree $4$ Conductor $1444$ Sign $1$ Analytic cond. $37.1439$ Root an. cond. $2.46872$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·2-s + 3·3-s + 48·4-s − 45·5-s − 24·6-s + 114·7-s − 256·8-s − 119·9-s + 360·10-s + 661·11-s + 144·12-s + 1.61e3·13-s − 912·14-s − 135·15-s + 1.28e3·16-s + 64·17-s + 952·18-s + 722·19-s − 2.16e3·20-s + 342·21-s − 5.28e3·22-s − 3.18e3·23-s − 768·24-s − 1.48e3·25-s − 1.29e4·26-s − 12·27-s + 5.47e3·28-s + ⋯
 L(s)  = 1 − 1.41·2-s + 0.192·3-s + 3/2·4-s − 0.804·5-s − 0.272·6-s + 0.879·7-s − 1.41·8-s − 0.489·9-s + 1.13·10-s + 1.64·11-s + 0.288·12-s + 2.64·13-s − 1.24·14-s − 0.154·15-s + 5/4·16-s + 0.0537·17-s + 0.692·18-s + 0.458·19-s − 1.20·20-s + 0.169·21-s − 2.32·22-s − 1.25·23-s − 0.272·24-s − 0.476·25-s − 3.74·26-s − 0.00316·27-s + 1.31·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1444$$    =    $$2^{2} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$37.1439$$ Root analytic conductor: $$2.46872$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1444,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.261890779$$ $$L(\frac12)$$ $$\approx$$ $$1.261890779$$ $$L(\frac{7}{2})$$ 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)$
bad2$C_1$ $$( 1 + p^{2} T )^{2}$$
19$C_1$ $$( 1 - p^{2} T )^{2}$$
good3$D_{4}$ $$1 - p T + 128 T^{2} - p^{6} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 + 9 p T + 3514 T^{2} + 9 p^{6} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 - 114 T + 31099 T^{2} - 114 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 661 T + 430972 T^{2} - 661 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 1613 T + 1384022 T^{2} - 1613 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 64 T + 2804713 T^{2} - 64 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 3185 T + 14543782 T^{2} + 3185 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 2481 T + 6815332 T^{2} + 2481 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 1180 T + 21633278 T^{2} + 1180 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 10488 T + 155529814 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 16630 T + 170295586 T^{2} - 16630 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 - 11303 T + 297510638 T^{2} - 11303 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 12155 T + 47754214 T^{2} + 12155 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 20585 T + 812203882 T^{2} - 20585 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 78581 T + 2971672216 T^{2} + 78581 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 7805 T - 748756690 T^{2} - 7805 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 62488 T + 4005427642 T^{2} + 62488 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 16218 T + 1004140843 T^{2} - 16218 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 67122 T + 7273984870 T^{2} - 67122 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 10714 T + 5751433246 T^{2} + 10714 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 128188 T + 8689285330 T^{2} - 128188 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 178558 T + 22668743394 T^{2} - 178558 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$