# Properties

 Label 2-1575-35.2-c0-0-1 Degree $2$ Conductor $1575$ Sign $0.997 + 0.0677i$ Analytic cond. $0.786027$ Root an. cond. $0.886581$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)4-s + (−0.965 − 0.258i)7-s + (1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (0.866 + 0.5i)19-s + (0.965 − 0.258i)28-s + (1 + 1.73i)31-s + (1.67 − 0.448i)37-s + (0.866 + 0.499i)49-s + (−0.448 + 1.67i)52-s + (0.5 − 0.866i)61-s + 0.999i·64-s + (0.448 − 1.67i)67-s + (−1.67 − 0.448i)73-s − 0.999·76-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)4-s + (−0.965 − 0.258i)7-s + (1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (0.866 + 0.5i)19-s + (0.965 − 0.258i)28-s + (1 + 1.73i)31-s + (1.67 − 0.448i)37-s + (0.866 + 0.499i)49-s + (−0.448 + 1.67i)52-s + (0.5 − 0.866i)61-s + 0.999i·64-s + (0.448 − 1.67i)67-s + (−1.67 − 0.448i)73-s − 0.999·76-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1575$$    =    $$3^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.997 + 0.0677i$ Analytic conductor: $$0.786027$$ Root analytic conductor: $$0.886581$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1575} (982, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1575,\ (\ :0),\ 0.997 + 0.0677i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8618631498$$ $$L(\frac12)$$ $$\approx$$ $$0.8618631498$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1 + (0.965 + 0.258i)T$$
good2 $$1 + (0.866 - 0.5i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T^{2}$$
13 $$1 + (-1.22 + 1.22i)T - iT^{2}$$
17 $$1 + (-0.866 - 0.5i)T^{2}$$
19 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 + (-0.866 + 0.5i)T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + (0.866 - 0.5i)T^{2}$$
53 $$1 + (0.866 + 0.5i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2}$$
79 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + (1.22 + 1.22i)T + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$