# Properties

 Label 2-35-7.6-c2-0-2 Degree $2$ Conductor $35$ Sign $0.958 + 0.285i$ 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 + 2·2-s − 2.23i·3-s + 2.23i·5-s − 4.47i·6-s + (−2 + 6.70i)7-s − 8·8-s + 3.99·9-s + 4.47i·10-s − 11-s − 20.1i·13-s + (−4 + 13.4i)14-s + 5.00·15-s − 16·16-s − 6.70i·17-s + 7.99·18-s + 13.4i·19-s + ⋯
 L(s)  = 1 + 2-s − 0.745i·3-s + 0.447i·5-s − 0.745i·6-s + (−0.285 + 0.958i)7-s − 8-s + 0.444·9-s + 0.447i·10-s − 0.0909·11-s − 1.54i·13-s + (−0.285 + 0.958i)14-s + 0.333·15-s − 16-s − 0.394i·17-s + 0.444·18-s + 0.706i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $0.958 + 0.285i$ 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: $$(2,\ 35,\ (\ :1),\ 0.958 + 0.285i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.40821 - 0.205456i$$ $$L(\frac12)$$ $$\approx$$ $$1.40821 - 0.205456i$$ $$L(2)$$ 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 + (2 - 6.70i)T$$
good2 $$1 - 2T + 4T^{2}$$
3 $$1 + 2.23iT - 9T^{2}$$
11 $$1 + T + 121T^{2}$$
13 $$1 + 20.1iT - 169T^{2}$$
17 $$1 + 6.70iT - 289T^{2}$$
19 $$1 - 13.4iT - 361T^{2}$$
23 $$1 - 8T + 529T^{2}$$
29 $$1 - 41T + 841T^{2}$$
31 $$1 - 40.2iT - 961T^{2}$$
37 $$1 + 28T + 1.36e3T^{2}$$
41 $$1 - 13.4iT - 1.68e3T^{2}$$
43 $$1 + 82T + 1.84e3T^{2}$$
47 $$1 - 20.1iT - 2.20e3T^{2}$$
53 $$1 - 74T + 2.80e3T^{2}$$
59 $$1 + 93.9iT - 3.48e3T^{2}$$
61 $$1 - 80.4iT - 3.72e3T^{2}$$
67 $$1 - 2T + 4.48e3T^{2}$$
71 $$1 - 14T + 5.04e3T^{2}$$
73 $$1 + 67.0iT - 5.32e3T^{2}$$
79 $$1 + 19T + 6.24e3T^{2}$$
83 $$1 - 93.9iT - 6.88e3T^{2}$$
89 $$1 - 107. iT - 7.92e3T^{2}$$
97 $$1 + 60.3iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$