# Properties

 Label 2-150-15.8-c1-0-4 Degree $2$ Conductor $150$ Sign $0.828 - 0.559i$ Analytic cond. $1.19775$ Root an. cond. $1.09442$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 + 0.707i)2-s + (1.67 − 0.448i)3-s + 1.00i·4-s + (1.5 + 0.866i)6-s + (−2.44 + 2.44i)7-s + (−0.707 + 0.707i)8-s + (2.59 − 1.50i)9-s − 5.19i·11-s + (0.448 + 1.67i)12-s − 3.46·14-s − 1.00·16-s + (−2.12 − 2.12i)17-s + (2.89 + 0.776i)18-s + i·19-s + (−3 + 5.19i)21-s + (3.67 − 3.67i)22-s + ⋯
 L(s)  = 1 + (0.499 + 0.499i)2-s + (0.965 − 0.258i)3-s + 0.500i·4-s + (0.612 + 0.353i)6-s + (−0.925 + 0.925i)7-s + (−0.250 + 0.250i)8-s + (0.866 − 0.5i)9-s − 1.56i·11-s + (0.129 + 0.482i)12-s − 0.925·14-s − 0.250·16-s + (−0.514 − 0.514i)17-s + (0.683 + 0.183i)18-s + 0.229i·19-s + (−0.654 + 1.13i)21-s + (0.783 − 0.783i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ Sign: $0.828 - 0.559i$ Analytic conductor: $$1.19775$$ Root analytic conductor: $$1.09442$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{150} (143, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 150,\ (\ :1/2),\ 0.828 - 0.559i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.63019 + 0.499193i$$ $$L(\frac12)$$ $$\approx$$ $$1.63019 + 0.499193i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.707 - 0.707i)T$$
3 $$1 + (-1.67 + 0.448i)T$$
5 $$1$$
good7 $$1 + (2.44 - 2.44i)T - 7iT^{2}$$
11 $$1 + 5.19iT - 11T^{2}$$
13 $$1 + 13iT^{2}$$
17 $$1 + (2.12 + 2.12i)T + 17iT^{2}$$
19 $$1 - iT - 19T^{2}$$
23 $$1 + (4.24 - 4.24i)T - 23iT^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 2T + 31T^{2}$$
37 $$1 + (-2.44 + 2.44i)T - 37iT^{2}$$
41 $$1 - 5.19iT - 41T^{2}$$
43 $$1 + (-2.44 - 2.44i)T + 43iT^{2}$$
47 $$1 + 47iT^{2}$$
53 $$1 + (-4.24 + 4.24i)T - 53iT^{2}$$
59 $$1 + 10.3T + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 + (3.67 - 3.67i)T - 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-6.12 - 6.12i)T + 73iT^{2}$$
79 $$1 - 14iT - 79T^{2}$$
83 $$1 + (-2.12 + 2.12i)T - 83iT^{2}$$
89 $$1 - 15.5T + 89T^{2}$$
97 $$1 + (4.89 - 4.89i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$