# Properties

 Degree $4$ Conductor $1444$ Sign $1$ Motivic weight $7$ Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·2-s − 69·3-s + 192·4-s + 155·5-s − 1.10e3·6-s − 2.23e3·7-s + 2.04e3·8-s + 621·9-s + 2.48e3·10-s − 3.29e3·11-s − 1.32e4·12-s − 1.34e4·13-s − 3.58e4·14-s − 1.06e4·15-s + 2.04e4·16-s − 3.22e4·17-s + 9.93e3·18-s − 1.37e4·19-s + 2.97e4·20-s + 1.54e5·21-s − 5.27e4·22-s − 8.25e4·23-s − 1.41e5·24-s + 1.38e4·25-s − 2.14e5·26-s + 9.19e4·27-s − 4.29e5·28-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.47·3-s + 3/2·4-s + 0.554·5-s − 2.08·6-s − 2.46·7-s + 1.41·8-s + 0.283·9-s + 0.784·10-s − 0.746·11-s − 2.21·12-s − 1.69·13-s − 3.48·14-s − 0.818·15-s + 5/4·16-s − 1.59·17-s + 0.401·18-s − 0.458·19-s + 0.831·20-s + 3.63·21-s − 1.05·22-s − 1.41·23-s − 2.08·24-s + 0.177·25-s − 2.39·26-s + 0.898·27-s − 3.69·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1444$$    =    $$2^{2} \cdot 19^{2}$$ Sign: $1$ Motivic weight: $$7$$ Character: induced by $\chi_{38} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1444,\ (\ :7/2, 7/2),\ 1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ 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$C_1$ $$( 1 - p^{3} T )^{2}$$
19$C_1$ $$( 1 + p^{3} T )^{2}$$
good3$D_{4}$ $$1 + 23 p T + 460 p^{2} T^{2} + 23 p^{8} T^{3} + p^{14} T^{4}$$
5$D_{4}$ $$1 - 31 p T + 10178 T^{2} - 31 p^{8} T^{3} + p^{14} T^{4}$$
7$D_{4}$ $$1 + 2238 T + 2775179 T^{2} + 2238 p^{7} T^{3} + p^{14} T^{4}$$
11$D_{4}$ $$1 + 3295 T + 38080340 T^{2} + 3295 p^{7} T^{3} + p^{14} T^{4}$$
13$D_{4}$ $$1 + 13427 T + 138664758 T^{2} + 13427 p^{7} T^{3} + p^{14} T^{4}$$
17$D_{4}$ $$1 + 32256 T + 1073810905 T^{2} + 32256 p^{7} T^{3} + p^{14} T^{4}$$
23$D_{4}$ $$1 + 82525 T + 5722043942 T^{2} + 82525 p^{7} T^{3} + p^{14} T^{4}$$
29$D_{4}$ $$1 + 12749 T + 39176144 p^{2} T^{2} + 12749 p^{7} T^{3} + p^{14} T^{4}$$
31$D_{4}$ $$1 - 258944 T + 64961539758 T^{2} - 258944 p^{7} T^{3} + p^{14} T^{4}$$
37$D_{4}$ $$1 + 149260 T + 183707435838 T^{2} + 149260 p^{7} T^{3} + p^{14} T^{4}$$
41$D_{4}$ $$1 - 339130 T - 4364095006 T^{2} - 339130 p^{7} T^{3} + p^{14} T^{4}$$
43$D_{4}$ $$1 + 83869 T + 6409629042 p T^{2} + 83869 p^{7} T^{3} + p^{14} T^{4}$$
47$D_{4}$ $$1 - 1471025 T + 1488141383726 T^{2} - 1471025 p^{7} T^{3} + p^{14} T^{4}$$
53$D_{4}$ $$1 + 945643 T + 2428814565902 T^{2} + 945643 p^{7} T^{3} + p^{14} T^{4}$$
59$D_{4}$ $$1 + 969009 T + 3139777837024 T^{2} + 969009 p^{7} T^{3} + p^{14} T^{4}$$
61$D_{4}$ $$1 + 1506755 T + 5925806441724 T^{2} + 1506755 p^{7} T^{3} + p^{14} T^{4}$$
67$D_{4}$ $$1 + 1848219 T + 11014380276458 T^{2} + 1848219 p^{7} T^{3} + p^{14} T^{4}$$
71$D_{4}$ $$1 + 3417184 T + 7686276501674 T^{2} + 3417184 p^{7} T^{3} + p^{14} T^{4}$$
73$D_{4}$ $$1 + 2499822 T + 17574764549507 T^{2} + 2499822 p^{7} T^{3} + p^{14} T^{4}$$
79$D_{4}$ $$1 - 2636926 T + 33245149452510 T^{2} - 2636926 p^{7} T^{3} + p^{14} T^{4}$$
83$D_{4}$ $$1 + 10059354 T + 77358130536910 T^{2} + 10059354 p^{7} T^{3} + p^{14} T^{4}$$
89$D_{4}$ $$1 + 3506160 T + 657141629522 p T^{2} + 3506160 p^{7} T^{3} + p^{14} T^{4}$$
97$D_{4}$ $$1 - 60758 p T + 73244272821042 T^{2} - 60758 p^{8} T^{3} + p^{14} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$