# Properties

 Label 2-150-3.2-c2-0-1 Degree $2$ Conductor $150$ Sign $0.235 - 0.971i$ Analytic cond. $4.08720$ Root an. cond. $2.02168$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s + (−2.91 − 0.707i)3-s − 2.00·4-s + (−1.00 + 4.12i)6-s − 5.83·7-s + 2.82i·8-s + (8 + 4.12i)9-s + 16.4i·11-s + (5.83 + 1.41i)12-s + 8.24i·14-s + 4.00·16-s + 11.3i·17-s + (5.83 − 11.3i)18-s − 12·19-s + (17 + 4.12i)21-s + 23.3·22-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.971 − 0.235i)3-s − 0.500·4-s + (−0.166 + 0.687i)6-s − 0.832·7-s + 0.353i·8-s + (0.888 + 0.458i)9-s + 1.49i·11-s + (0.485 + 0.117i)12-s + 0.589i·14-s + 0.250·16-s + 0.665i·17-s + (0.323 − 0.628i)18-s − 0.631·19-s + (0.809 + 0.196i)21-s + 1.06·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ Sign: $0.235 - 0.971i$ Analytic conductor: $$4.08720$$ Root analytic conductor: $$2.02168$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{150} (101, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 150,\ (\ :1),\ 0.235 - 0.971i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.337381 + 0.265335i$$ $$L(\frac12)$$ $$\approx$$ $$0.337381 + 0.265335i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41iT$$
3 $$1 + (2.91 + 0.707i)T$$
5 $$1$$
good7 $$1 + 5.83T + 49T^{2}$$
11 $$1 - 16.4iT - 121T^{2}$$
13 $$1 + 169T^{2}$$
17 $$1 - 11.3iT - 289T^{2}$$
19 $$1 + 12T + 361T^{2}$$
23 $$1 - 24.0iT - 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 + 32T + 961T^{2}$$
37 $$1 + 23.3T + 1.36e3T^{2}$$
41 $$1 + 57.7iT - 1.68e3T^{2}$$
43 $$1 + 40.8T + 1.84e3T^{2}$$
47 $$1 - 35.3iT - 2.20e3T^{2}$$
53 $$1 - 67.8iT - 2.80e3T^{2}$$
59 $$1 + 16.4iT - 3.48e3T^{2}$$
61 $$1 + 16T + 3.72e3T^{2}$$
67 $$1 - 5.83T + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 116.T + 5.32e3T^{2}$$
79 $$1 - 72T + 6.24e3T^{2}$$
83 $$1 + 43.8iT - 6.88e3T^{2}$$
89 $$1 - 65.9iT - 7.92e3T^{2}$$
97 $$1 - 163.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$