# Properties

 Label 2-35-35.34-c4-0-10 Degree $2$ Conductor $35$ Sign $-0.542 + 0.839i$ Analytic cond. $3.61794$ Root an. cond. $1.90209$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.44i·2-s − 5·3-s + 10·4-s + (−5 − 24.4i)5-s + 12.2i·6-s + (−35 − 34.2i)7-s − 63.6i·8-s − 56·9-s + (−59.9 + 12.2i)10-s + 89·11-s − 50·12-s + 5·13-s + (−84 + 85.7i)14-s + (25 + 122. i)15-s + 4.00·16-s + 485·17-s + ⋯
 L(s)  = 1 − 0.612i·2-s − 0.555·3-s + 0.625·4-s + (−0.200 − 0.979i)5-s + 0.340i·6-s + (−0.714 − 0.699i)7-s − 0.995i·8-s − 0.691·9-s + (−0.599 + 0.122i)10-s + 0.735·11-s − 0.347·12-s + 0.0295·13-s + (−0.428 + 0.437i)14-s + (0.111 + 0.544i)15-s + 0.0156·16-s + 1.67·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $-0.542 + 0.839i$ Analytic conductor: $$3.61794$$ Root analytic conductor: $$1.90209$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{35} (34, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 35,\ (\ :2),\ -0.542 + 0.839i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.564677 - 1.03737i$$ $$L(\frac12)$$ $$\approx$$ $$0.564677 - 1.03737i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (5 + 24.4i)T$$
7 $$1 + (35 + 34.2i)T$$
good2 $$1 + 2.44iT - 16T^{2}$$
3 $$1 + 5T + 81T^{2}$$
11 $$1 - 89T + 1.46e4T^{2}$$
13 $$1 - 5T + 2.85e4T^{2}$$
17 $$1 - 485T + 8.35e4T^{2}$$
19 $$1 + 220. iT - 1.30e5T^{2}$$
23 $$1 - 700. iT - 2.79e5T^{2}$$
29 $$1 - 191T + 7.07e5T^{2}$$
31 $$1 - 1.05e3iT - 9.23e5T^{2}$$
37 $$1 + 1.63e3iT - 1.87e6T^{2}$$
41 $$1 + 2.91e3iT - 2.82e6T^{2}$$
43 $$1 + 377. iT - 3.41e6T^{2}$$
47 $$1 - 2.19e3T + 4.87e6T^{2}$$
53 $$1 - 1.58e3iT - 7.89e6T^{2}$$
59 $$1 - 3.62e3iT - 1.21e7T^{2}$$
61 $$1 + 1.93e3iT - 1.38e7T^{2}$$
67 $$1 + 2.04e3iT - 2.01e7T^{2}$$
71 $$1 - 4.45e3T + 2.54e7T^{2}$$
73 $$1 + 8.65e3T + 2.83e7T^{2}$$
79 $$1 - 5.56e3T + 3.89e7T^{2}$$
83 $$1 + 1.99e3T + 4.74e7T^{2}$$
89 $$1 - 808. iT - 6.27e7T^{2}$$
97 $$1 + 9.23e3T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$