# Properties

 Label 2-2520-2520.2477-c0-0-1 Degree $2$ Conductor $2520$ Sign $-0.999 - 0.0136i$ Analytic cond. $1.25764$ Root an. cond. $1.12144$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.258 + 0.965i)2-s + (−0.923 + 0.382i)3-s + (−0.866 + 0.499i)4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.991 + 0.130i)10-s + (0.608 − 0.793i)12-s + (0.739 + 0.198i)13-s + (0.499 + 0.866i)14-s + (−0.258 − 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s + 1.58i·19-s + ⋯
 L(s)  = 1 + (0.258 + 0.965i)2-s + (−0.923 + 0.382i)3-s + (−0.866 + 0.499i)4-s + (−0.130 + 0.991i)5-s + (−0.608 − 0.793i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.991 + 0.130i)10-s + (0.608 − 0.793i)12-s + (0.739 + 0.198i)13-s + (0.499 + 0.866i)14-s + (−0.258 − 0.965i)15-s + (0.500 − 0.866i)16-s + (0.866 + 0.5i)18-s + 1.58i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2520$$    =    $$2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Sign: $-0.999 - 0.0136i$ Analytic conductor: $$1.25764$$ Root analytic conductor: $$1.12144$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2520} (2477, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2520,\ (\ :0),\ -0.999 - 0.0136i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9111852445$$ $$L(\frac12)$$ $$\approx$$ $$0.9111852445$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.258 - 0.965i)T$$
3 $$1 + (0.923 - 0.382i)T$$
5 $$1 + (0.130 - 0.991i)T$$
7 $$1 + (-0.965 + 0.258i)T$$
good11 $$1 + (-0.5 - 0.866i)T^{2}$$
13 $$1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - 1.58iT - T^{2}$$
23 $$1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2}$$
29 $$1 + (0.5 + 0.866i)T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.866 - 0.5i)T^{2}$$
47 $$1 + (0.866 - 0.5i)T^{2}$$
53 $$1 - iT^{2}$$
59 $$1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.866 + 0.5i)T^{2}$$
71 $$1 - 0.517iT - T^{2}$$
73 $$1 + iT^{2}$$
79 $$1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2}$$
89 $$1 - T^{2}$$
97 $$1 + (-0.866 + 0.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$