# Properties

 Label 2-120-120.77-c1-0-10 Degree $2$ Conductor $120$ Sign $0.793 - 0.608i$ Analytic cond. $0.958204$ Root an. cond. $0.978879$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (0.224 + 2.22i)5-s + 2.44·6-s + (−3.44 − 3.44i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−2 + 2.44i)10-s + 1.55·11-s + (2.44 + 2.44i)12-s − 6.89i·14-s + (2.99 + 2.44i)15-s − 4·16-s + (2.99 − 2.99i)18-s + ⋯
 L(s)  = 1 + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s + (0.100 + 0.994i)5-s + 0.999·6-s + (−1.30 − 1.30i)7-s + (−0.707 + 0.707i)8-s − 0.999i·9-s + (−0.632 + 0.774i)10-s + 0.467·11-s + (0.707 + 0.707i)12-s − 1.84i·14-s + (0.774 + 0.632i)15-s − 16-s + (0.707 − 0.707i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$120$$    =    $$2^{3} \cdot 3 \cdot 5$$ Sign: $0.793 - 0.608i$ Analytic conductor: $$0.958204$$ Root analytic conductor: $$0.978879$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{120} (77, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 120,\ (\ :1/2),\ 0.793 - 0.608i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.52737 + 0.518265i$$ $$L(\frac12)$$ $$\approx$$ $$1.52737 + 0.518265i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 - i)T$$
3 $$1 + (-1.22 + 1.22i)T$$
5 $$1 + (-0.224 - 2.22i)T$$
good7 $$1 + (3.44 + 3.44i)T + 7iT^{2}$$
11 $$1 - 1.55T + 11T^{2}$$
13 $$1 + 13iT^{2}$$
17 $$1 - 17iT^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 - 5.34iT - 29T^{2}$$
31 $$1 - 4.89T + 31T^{2}$$
37 $$1 - 37iT^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + 43iT^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + (-2.44 + 2.44i)T - 53iT^{2}$$
59 $$1 - 15.3iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-11.8 + 11.8i)T - 73iT^{2}$$
79 $$1 + 14.6iT - 79T^{2}$$
83 $$1 + (4 - 4i)T - 83iT^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 + (-8.79 - 8.79i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$