# Properties

 Label 2-5610-1.1-c1-0-82 Degree $2$ Conductor $5610$ Sign $-1$ Analytic cond. $44.7960$ Root an. cond. $6.69298$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s + 2·13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s − 22-s − 4·23-s + 24-s + 25-s − 2·26-s − 27-s − 6·29-s + 30-s − 8·31-s − 32-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5610$$    =    $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Sign: $-1$ Analytic conductor: $$44.7960$$ Root analytic conductor: $$6.69298$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 5610,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
5 $$1 - T$$
11 $$1 - T$$
17 $$1 + T$$
good7 $$1 + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + 4 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 2 T + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$