# Properties

 Label 2-1260-105.74-c0-0-1 Degree $2$ Conductor $1260$ Sign $0.807 - 0.589i$ Analytic cond. $0.628821$ Root an. cond. $0.792982$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.965 + 0.258i)5-s + (−0.866 + 0.5i)7-s + i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (0.866 + 0.499i)25-s − 1.41i·29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.866 + 0.5i)37-s − i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (1.22 + 0.707i)59-s + ⋯
 L(s)  = 1 + (0.965 + 0.258i)5-s + (−0.866 + 0.5i)7-s + i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (0.866 + 0.499i)25-s − 1.41i·29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.866 + 0.5i)37-s − i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (1.22 + 0.707i)59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1260$$    =    $$2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Sign: $0.807 - 0.589i$ Analytic conductor: $$0.628821$$ Root analytic conductor: $$0.792982$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1260} (809, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1260,\ (\ :0),\ 0.807 - 0.589i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.168101676$$ $$L(\frac12)$$ $$\approx$$ $$1.168101676$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (-0.965 - 0.258i)T$$
7 $$1 + (0.866 - 0.5i)T$$
good11 $$1 + (0.5 + 0.866i)T^{2}$$
13 $$1 - iT - T^{2}$$
17 $$1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
23 $$1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}$$
29 $$1 + 1.41iT - T^{2}$$
31 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
37 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + iT - T^{2}$$
47 $$1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}$$
53 $$1 + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
79 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$