# Properties

 Label 2-1950-5.4-c1-0-2 Degree $2$ Conductor $1950$ Sign $0.447 - 0.894i$ Analytic cond. $15.5708$ Root an. cond. $3.94598$ Motivic weight $1$ 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 − 4-s − 6-s + i·7-s + i·8-s − 9-s − 5·11-s + i·12-s − i·13-s + 14-s + 16-s − 5i·17-s + i·18-s + 21-s + 5i·22-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.50·11-s + 0.288i·12-s − 0.277i·13-s + 0.267·14-s + 0.250·16-s − 1.21i·17-s + 0.235i·18-s + 0.218·21-s + 1.06i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1950$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 13$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$15.5708$$ Root analytic conductor: $$3.94598$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1950} (1249, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1950,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3982971107$$ $$L(\frac12)$$ $$\approx$$ $$0.3982971107$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + iT$$
5 $$1$$
13 $$1 + iT$$
good7 $$1 - iT - 7T^{2}$$
11 $$1 + 5T + 11T^{2}$$
17 $$1 + 5iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 7T + 29T^{2}$$
31 $$1 + 9T + 31T^{2}$$
37 $$1 - 8iT - 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 - 9iT - 47T^{2}$$
53 $$1 - 11iT - 53T^{2}$$
59 $$1 + T + 59T^{2}$$
61 $$1 + 7T + 61T^{2}$$
67 $$1 - 15iT - 67T^{2}$$
71 $$1 + 8T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 9iT - 83T^{2}$$
89 $$1 + 16T + 89T^{2}$$
97 $$1 + 2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$