# Properties

 Label 2-2175-435.434-c0-0-10 Degree $2$ Conductor $2175$ Sign $-0.894 + 0.447i$ Analytic cond. $1.08546$ Root an. cond. $1.04185$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯
 L(s)  = 1 − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2175$$    =    $$3 \cdot 5^{2} \cdot 29$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$1.08546$$ Root analytic conductor: $$1.04185$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2175} (2174, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2175,\ (\ :0),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.299551278$$ $$L(\frac12)$$ $$\approx$$ $$1.299551278$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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