# Properties

 Label 2-1045-1045.417-c0-0-3 Degree $2$ Conductor $1045$ Sign $0.525 + 0.850i$ Analytic cond. $0.521522$ Root an. cond. $0.722165$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·4-s + 5-s + (1 − i)7-s − i·9-s + i·11-s − 16-s + (−1 + i)17-s − 19-s − i·20-s + (−1 + i)23-s + 25-s + (−1 − i)28-s + (1 − i)35-s − 36-s + (1 + i)43-s + 44-s + ⋯
 L(s)  = 1 − i·4-s + 5-s + (1 − i)7-s − i·9-s + i·11-s − 16-s + (−1 + i)17-s − 19-s − i·20-s + (−1 + i)23-s + 25-s + (−1 − i)28-s + (1 − i)35-s − 36-s + (1 + i)43-s + 44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $0.525 + 0.850i$ Analytic conductor: $$0.521522$$ Root analytic conductor: $$0.722165$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (417, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1045,\ (\ :0),\ 0.525 + 0.850i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.257709130$$ $$L(\frac12)$$ $$\approx$$ $$1.257709130$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - T$$
11 $$1 - iT$$
19 $$1 + T$$
good2 $$1 + iT^{2}$$
3 $$1 + iT^{2}$$
7 $$1 + (-1 + i)T - iT^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + (1 - i)T - iT^{2}$$
23 $$1 + (1 - i)T - iT^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1 - i)T + iT^{2}$$
47 $$1 + (-1 - i)T + iT^{2}$$
53 $$1 + iT^{2}$$
59 $$1 + T^{2}$$
61 $$1 + 2iT - T^{2}$$
67 $$1 - iT^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (-1 - i)T + iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 + (1 + i)T + iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 - iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$