# Properties

 Label 2-1710-5.4-c1-0-30 Degree $2$ Conductor $1710$ Sign $-0.447 + 0.894i$ Analytic cond. $13.6544$ Root an. cond. $3.69518$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s + (−2 − i)5-s + i·8-s + (−1 + 2i)10-s + 4·11-s + 4i·13-s + 16-s − 6i·17-s + 19-s + (2 + i)20-s − 4i·22-s + (3 + 4i)25-s + 4·26-s − 2·29-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.353i·8-s + (−0.316 + 0.632i)10-s + 1.20·11-s + 1.10i·13-s + 0.250·16-s − 1.45i·17-s + 0.229·19-s + (0.447 + 0.223i)20-s − 0.852i·22-s + (0.600 + 0.800i)25-s + 0.784·26-s − 0.371·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1710$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 19$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$13.6544$$ Root analytic conductor: $$3.69518$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1710} (1369, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1710,\ (\ :1/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.255632201$$ $$L(\frac12)$$ $$\approx$$ $$1.255632201$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1$$
5 $$1 + (2 + i)T$$
19 $$1 - T$$
good7 $$1 - 7T^{2}$$
11 $$1 - 4T + 11T^{2}$$
13 $$1 - 4iT - 13T^{2}$$
17 $$1 + 6iT - 17T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 4iT - 37T^{2}$$
41 $$1 - 12T + 41T^{2}$$
43 $$1 + 6iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 14iT - 53T^{2}$$
59 $$1 + 10T + 59T^{2}$$
61 $$1 + 6T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + 8iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 8T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$