# Properties

 Label 4-1216e2-1.1-c3e2-0-6 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 − 3-s + 5·5-s + 6·7-s − 39·9-s − 15·11-s + 59·13-s − 5·15-s − 104·17-s − 38·19-s − 6·21-s + 21·23-s − 217·25-s + 52·27-s + 137·29-s − 4·31-s + 15·33-s + 30·35-s + 152·37-s − 59·39-s − 210·41-s + 67·43-s − 195·45-s − 273·47-s − 431·49-s + 104·51-s − 209·53-s − 75·55-s + ⋯
 L(s)  = 1 − 0.192·3-s + 0.447·5-s + 0.323·7-s − 1.44·9-s − 0.411·11-s + 1.25·13-s − 0.0860·15-s − 1.48·17-s − 0.458·19-s − 0.0623·21-s + 0.190·23-s − 1.73·25-s + 0.370·27-s + 0.877·29-s − 0.0231·31-s + 0.0791·33-s + 0.144·35-s + 0.675·37-s − 0.242·39-s − 0.799·41-s + 0.237·43-s − 0.645·45-s − 0.847·47-s − 1.25·49-s + 0.285·51-s − 0.541·53-s − 0.183·55-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$D_{4}$ $$1 + T + 40 T^{2} + p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 - p T + 242 T^{2} - p^{4} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 - 6 T + 467 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 15 T + 2020 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 59 T + 4566 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 104 T + 11105 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 21 T + 8926 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 137 T + 1804 p T^{2} - 137 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 4 T + 48414 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 152 T + 95910 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 210 T + 33442 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 67 T + 152598 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 273 T + 197422 T^{2} + 273 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 209 T + 308546 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 799 T + 513800 T^{2} - 799 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 149 T + 419484 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 3 p T + 611270 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 792 T + 730138 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 246 T + 719291 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 254 T + 817014 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 374 T + 840590 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 564 T + 942034 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 178 T - 159966 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$