# Properties

 Label 2-35-7.6-c2-0-3 Degree $2$ Conductor $35$ Sign $i$ Analytic cond. $0.953680$ Root an. cond. $0.976565$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 4.47i·3-s − 3·4-s − 2.23i·5-s + 4.47i·6-s + 7·7-s + 7·8-s − 11.0·9-s + 2.23i·10-s + 2·11-s + 13.4i·12-s + 13.4i·13-s − 7·14-s − 10.0·15-s + 5·16-s − 26.8i·17-s + ⋯
 L(s)  = 1 − 0.5·2-s − 1.49i·3-s − 0.750·4-s − 0.447i·5-s + 0.745i·6-s + 7-s + 0.875·8-s − 1.22·9-s + 0.223i·10-s + 0.181·11-s + 1.11i·12-s + 1.03i·13-s − 0.5·14-s − 0.666·15-s + 0.312·16-s − 1.57i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $i$ Analytic conductor: $$0.953680$$ Root analytic conductor: $$0.976565$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{35} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 35,\ (\ :1),\ i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.533518 - 0.533518i$$ $$L(\frac12)$$ $$\approx$$ $$0.533518 - 0.533518i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + 2.23iT$$
7 $$1 - 7T$$
good2 $$1 + T + 4T^{2}$$
3 $$1 + 4.47iT - 9T^{2}$$
11 $$1 - 2T + 121T^{2}$$
13 $$1 - 13.4iT - 169T^{2}$$
17 $$1 + 26.8iT - 289T^{2}$$
19 $$1 - 13.4iT - 361T^{2}$$
23 $$1 - 26T + 529T^{2}$$
29 $$1 + 22T + 841T^{2}$$
31 $$1 - 53.6iT - 961T^{2}$$
37 $$1 - 14T + 1.36e3T^{2}$$
41 $$1 - 26.8iT - 1.68e3T^{2}$$
43 $$1 + 34T + 1.84e3T^{2}$$
47 $$1 - 26.8iT - 2.20e3T^{2}$$
53 $$1 + 34T + 2.80e3T^{2}$$
59 $$1 - 40.2iT - 3.48e3T^{2}$$
61 $$1 + 93.9iT - 3.72e3T^{2}$$
67 $$1 - 14T + 4.48e3T^{2}$$
71 $$1 - 62T + 5.04e3T^{2}$$
73 $$1 - 53.6iT - 5.32e3T^{2}$$
79 $$1 - 38T + 6.24e3T^{2}$$
83 $$1 + 40.2iT - 6.88e3T^{2}$$
89 $$1 - 26.8iT - 7.92e3T^{2}$$
97 $$1 - 26.8iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$