# Properties

 Label 4-1840e2-1.1-c3e2-0-1 Degree $4$ Conductor $3385600$ Sign $1$ Analytic cond. $11786.0$ Root an. cond. $10.4193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·3-s + 10·5-s − 7-s − 20·9-s + 27·11-s − 15·13-s + 30·15-s − 79·17-s + 71·19-s − 3·21-s + 46·23-s + 75·25-s − 66·27-s − 430·29-s + 305·31-s + 81·33-s − 10·35-s − 68·37-s − 45·39-s − 593·41-s − 648·43-s − 200·45-s − 382·47-s − 4·49-s − 237·51-s − 464·53-s + 270·55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.894·5-s − 0.0539·7-s − 0.740·9-s + 0.740·11-s − 0.320·13-s + 0.516·15-s − 1.12·17-s + 0.857·19-s − 0.0311·21-s + 0.417·23-s + 3/5·25-s − 0.470·27-s − 2.75·29-s + 1.76·31-s + 0.427·33-s − 0.0482·35-s − 0.302·37-s − 0.184·39-s − 2.25·41-s − 2.29·43-s − 0.662·45-s − 1.18·47-s − 0.0116·49-s − 0.650·51-s − 1.20·53-s + 0.661·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3385600$$    =    $$2^{8} \cdot 5^{2} \cdot 23^{2}$$ Sign: $1$ Analytic conductor: $$11786.0$$ Root analytic conductor: $$10.4193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 3385600,\ (\ :3/2, 3/2),\ 1)$$

## 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 $$1$$
5$C_1$ $$( 1 - p T )^{2}$$
23$C_1$ $$( 1 - p T )^{2}$$
good3$D_{4}$ $$1 - p T + 29 T^{2} - p^{4} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 + T + 5 T^{2} + p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 27 T + 2163 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 15 T + 4205 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 79 T + 10051 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 71 T + 10373 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 430 T + 3182 p T^{2} + 430 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 305 T + 74963 T^{2} - 305 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 68 T + 81098 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 593 T + 222457 T^{2} + 593 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 648 T + 262246 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 382 T + 94906 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 464 T + 298822 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 18 T + 331378 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 7 T + 365439 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 60 T + 445030 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 1029 T + 724137 T^{2} - 1029 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 74 T + 674654 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 692 T + 299630 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 1460 T + 1111418 T^{2} - 1460 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 220 T + 539138 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 1339 T + 1150631 T^{2} - 1339 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$