# Properties

 Label 4-1216e2-1.1-c3e2-0-8 Degree $4$ Conductor $1478656$ Sign $1$ Analytic cond. $5147.53$ Root an. cond. $8.47032$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5·3-s + 5·5-s − 30·7-s − 27·9-s + 71·11-s + 35·13-s + 25·15-s + 38·19-s − 150·21-s − 5·23-s − 25·25-s − 260·27-s − 155·29-s − 88·31-s + 355·33-s − 150·35-s − 380·37-s + 175·39-s − 142·41-s − 155·43-s − 135·45-s − 455·47-s + 121·49-s + 275·53-s + 355·55-s + 190·57-s + 873·59-s + ⋯
 L(s)  = 1 + 0.962·3-s + 0.447·5-s − 1.61·7-s − 9-s + 1.94·11-s + 0.746·13-s + 0.430·15-s + 0.458·19-s − 1.55·21-s − 0.0453·23-s − 1/5·25-s − 1.85·27-s − 0.992·29-s − 0.509·31-s + 1.87·33-s − 0.724·35-s − 1.68·37-s + 0.718·39-s − 0.540·41-s − 0.549·43-s − 0.447·45-s − 1.41·47-s + 0.352·49-s + 0.712·53-s + 0.870·55-s + 0.441·57-s + 1.92·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1478656$$    =    $$2^{12} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$5147.53$$ Root analytic conductor: $$8.47032$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1478656,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 $$1$$
19$C_1$ $$( 1 - p T )^{2}$$
good3$D_{4}$ $$1 - 5 T + 52 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 - p T + 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 71 T + 3716 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 35 T + 3702 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4}$$
17$C_2^2$ $$1 + 9529 T^{2} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 5 T - 4378 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 380 T + 136878 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 155 T + 52086 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 455 T + 255038 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 275 T + 191846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 445 T + 250812 T^{2} + 445 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 645 T + 674834 T^{2} + 645 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 1712 T + 1445258 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 1274 T + 1391022 T^{2} + 1274 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 90 T + 1038382 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$