# Properties

 Label 2-1900-95.94-c2-0-48 Degree $2$ Conductor $1900$ Sign $-0.447 + 0.894i$ Analytic cond. $51.7712$ Root an. cond. $7.19522$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8.82i·7-s − 9·9-s + 17.3·11-s − 33.9i·17-s + 19·19-s + 30i·23-s − 31.1i·43-s + 11.5i·47-s − 28.8·49-s − 108.·61-s + 79.4i·63-s − 137. i·73-s − 153. i·77-s + 81·81-s − 90i·83-s + ⋯
 L(s)  = 1 − 1.26i·7-s − 9-s + 1.57·11-s − 1.99i·17-s + 19-s + 1.30i·23-s − 0.725i·43-s + 0.246i·47-s − 0.589·49-s − 1.77·61-s + 1.26i·63-s − 1.87i·73-s − 1.99i·77-s + 81-s − 1.08i·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1900$$    =    $$2^{2} \cdot 5^{2} \cdot 19$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$51.7712$$ Root analytic conductor: $$7.19522$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1900} (949, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1900,\ (\ :1),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.637992078$$ $$L(\frac12)$$ $$\approx$$ $$1.637992078$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
19 $$1 - 19T$$
good3 $$1 + 9T^{2}$$
7 $$1 + 8.82iT - 49T^{2}$$
11 $$1 - 17.3T + 121T^{2}$$
13 $$1 + 169T^{2}$$
17 $$1 + 33.9iT - 289T^{2}$$
23 $$1 - 30iT - 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 + 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 + 31.1iT - 1.84e3T^{2}$$
47 $$1 - 11.5iT - 2.20e3T^{2}$$
53 $$1 + 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 + 108.T + 3.72e3T^{2}$$
67 $$1 + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 137. iT - 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 + 90iT - 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$