# Properties

 Label 4-38e2-1.1-c3e2-0-3 Degree $4$ Conductor $1444$ Sign $1$ Analytic cond. $5.02688$ Root an. cond. $1.49735$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 9·3-s + 12·4-s − 9·5-s + 36·6-s − 18·7-s + 32·8-s + 25·9-s − 36·10-s − 17·11-s + 108·12-s + 17·13-s − 72·14-s − 81·15-s + 80·16-s − 80·17-s + 100·18-s + 38·19-s − 108·20-s − 162·21-s − 68·22-s + 73·23-s + 288·24-s − 25·25-s + 68·26-s − 36·27-s − 216·28-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.804·5-s + 2.44·6-s − 0.971·7-s + 1.41·8-s + 0.925·9-s − 1.13·10-s − 0.465·11-s + 2.59·12-s + 0.362·13-s − 1.37·14-s − 1.39·15-s + 5/4·16-s − 1.14·17-s + 1.30·18-s + 0.458·19-s − 1.20·20-s − 1.68·21-s − 0.658·22-s + 0.661·23-s + 2.44·24-s − 1/5·25-s + 0.512·26-s − 0.256·27-s − 1.45·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1444$$    =    $$2^{2} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$5.02688$$ Root analytic conductor: $$1.49735$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1444,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$4.303528832$$ $$L(\frac12)$$ $$\approx$$ $$4.303528832$$ $$L(\frac{5}{2})$$ 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)$
bad2$C_1$ $$( 1 - p T )^{2}$$
19$C_1$ $$( 1 - p T )^{2}$$
good3$D_{4}$ $$1 - p^{2} T + 56 T^{2} - p^{5} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 + 9 T + 106 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 + 18 T + 475 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 17 T + 2716 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 17 T + 1382 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 73 T + 22582 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 3 T + 40732 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 212 T + 35486 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 192 T + 96214 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 50 T + 136642 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 677 T + 272702 T^{2} - 677 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 389 T + 153478 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 23 p T + 663970 T^{2} + 23 p^{4} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 287 T + 419944 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 - 313 T + 253596 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 1223 T + 867254 T^{2} - 1223 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 200 T + 480250 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 1350 T + 1380310 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 670 T + 568942 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 236 T + 778834 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 1294 T + 2054082 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$