# Properties

 Label 2-1440-1.1-c3-0-44 Degree $2$ Conductor $1440$ Sign $-1$ Analytic cond. $84.9627$ Root an. cond. $9.21752$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 5·5-s + 12·7-s − 24·11-s + 38·13-s + 6·17-s − 104·19-s + 100·23-s + 25·25-s − 230·29-s + 56·31-s − 60·35-s + 190·37-s − 202·41-s + 148·43-s + 124·47-s − 199·49-s − 206·53-s + 120·55-s − 128·59-s + 190·61-s − 190·65-s + 204·67-s − 440·71-s + 1.21e3·73-s − 288·77-s − 816·79-s − 1.41e3·83-s + ⋯
 L(s)  = 1 − 0.447·5-s + 0.647·7-s − 0.657·11-s + 0.810·13-s + 0.0856·17-s − 1.25·19-s + 0.906·23-s + 1/5·25-s − 1.47·29-s + 0.324·31-s − 0.289·35-s + 0.844·37-s − 0.769·41-s + 0.524·43-s + 0.384·47-s − 0.580·49-s − 0.533·53-s + 0.294·55-s − 0.282·59-s + 0.398·61-s − 0.362·65-s + 0.371·67-s − 0.735·71-s + 1.93·73-s − 0.426·77-s − 1.16·79-s − 1.86·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1440$$    =    $$2^{5} \cdot 3^{2} \cdot 5$$ Sign: $-1$ Analytic conductor: $$84.9627$$ Root analytic conductor: $$9.21752$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1440,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + p T$$
good7 $$1 - 12 T + p^{3} T^{2}$$
11 $$1 + 24 T + p^{3} T^{2}$$
13 $$1 - 38 T + p^{3} T^{2}$$
17 $$1 - 6 T + p^{3} T^{2}$$
19 $$1 + 104 T + p^{3} T^{2}$$
23 $$1 - 100 T + p^{3} T^{2}$$
29 $$1 + 230 T + p^{3} T^{2}$$
31 $$1 - 56 T + p^{3} T^{2}$$
37 $$1 - 190 T + p^{3} T^{2}$$
41 $$1 + 202 T + p^{3} T^{2}$$
43 $$1 - 148 T + p^{3} T^{2}$$
47 $$1 - 124 T + p^{3} T^{2}$$
53 $$1 + 206 T + p^{3} T^{2}$$
59 $$1 + 128 T + p^{3} T^{2}$$
61 $$1 - 190 T + p^{3} T^{2}$$
67 $$1 - 204 T + p^{3} T^{2}$$
71 $$1 + 440 T + p^{3} T^{2}$$
73 $$1 - 1210 T + p^{3} T^{2}$$
79 $$1 + 816 T + p^{3} T^{2}$$
83 $$1 + 1412 T + p^{3} T^{2}$$
89 $$1 - 214 T + p^{3} T^{2}$$
97 $$1 - 1202 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$