# Properties

 Degree $2$ Conductor $1734$ Sign $0.638 + 0.769i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + (0.707 − 0.707i)3-s − 4-s + (1.41 − 1.41i)5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s + (1.41 + 1.41i)10-s + (−2.82 − 2.82i)11-s + (−0.707 + 0.707i)12-s + 2·13-s − 2.00i·15-s + 16-s + 1.00·18-s + 4i·19-s + (−1.41 + 1.41i)20-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.632 − 0.632i)5-s + (0.288 + 0.288i)6-s − 0.353i·8-s − 0.333i·9-s + (0.447 + 0.447i)10-s + (−0.852 − 0.852i)11-s + (−0.204 + 0.204i)12-s + 0.554·13-s − 0.516i·15-s + 0.250·16-s + 0.235·18-s + 0.917i·19-s + (−0.316 + 0.316i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1734$$    =    $$2 \cdot 3 \cdot 17^{2}$$ Sign: $0.638 + 0.769i$ Motivic weight: $$1$$ Character: $\chi_{1734} (1483, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1734,\ (\ :1/2),\ 0.638 + 0.769i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.875569103$$ $$L(\frac12)$$ $$\approx$$ $$1.875569103$$ $$L(\frac{3}{2})$$ 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 + (-0.707 + 0.707i)T$$
17 $$1$$
good5 $$1 + (-1.41 + 1.41i)T - 5iT^{2}$$
7 $$1 + 7iT^{2}$$
11 $$1 + (2.82 + 2.82i)T + 11iT^{2}$$
13 $$1 - 2T + 13T^{2}$$
19 $$1 - 4iT - 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 + (-7.07 + 7.07i)T - 29iT^{2}$$
31 $$1 + (-5.65 + 5.65i)T - 31iT^{2}$$
37 $$1 + (1.41 - 1.41i)T - 37iT^{2}$$
41 $$1 + (7.07 + 7.07i)T + 41iT^{2}$$
43 $$1 + 12iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + 12iT - 59T^{2}$$
61 $$1 + (-7.07 - 7.07i)T + 61iT^{2}$$
67 $$1 + 12T + 67T^{2}$$
71 $$1 - 71iT^{2}$$
73 $$1 + (7.07 - 7.07i)T - 73iT^{2}$$
79 $$1 + (5.65 + 5.65i)T + 79iT^{2}$$
83 $$1 - 4iT - 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 + (-9.89 + 9.89i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$