# Properties

 Label 2-690-1.1-c1-0-10 Degree $2$ Conductor $690$ Sign $-1$ Analytic cond. $5.50967$ Root an. cond. $2.34727$ 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 − 2·7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 18-s − 4·19-s − 20-s + 2·21-s − 6·22-s − 23-s + 24-s + 25-s + 2·26-s − 27-s − 2·28-s − 2·29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.436·21-s − 1.27·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 0.371·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $-1$ Analytic conductor: $$5.50967$$ Root analytic conductor: $$2.34727$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{690} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 690,\ (\ :1/2),\ -1)$$

## 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$$
23 $$1 + T$$
good7 $$1 + 2 T + p T^{2}$$
11 $$1 - 6 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 14 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + 16 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}$$