# Properties

 Label 2-1950-65.64-c1-0-6 Degree $2$ Conductor $1950$ Sign $-0.581 - 0.813i$ 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 − 2-s + i·3-s + 4-s − i·6-s + 3.12·7-s − 8-s − 9-s + 5.12i·11-s + i·12-s + (0.561 + 3.56i)13-s − 3.12·14-s + 16-s − 2i·17-s + 18-s + 6i·19-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 1.18·7-s − 0.353·8-s − 0.333·9-s + 1.54i·11-s + 0.288i·12-s + (0.155 + 0.987i)13-s − 0.834·14-s + 0.250·16-s − 0.485i·17-s + 0.235·18-s + 1.37i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1950$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 13$$ Sign: $-0.581 - 0.813i$ Analytic conductor: $$15.5708$$ Root analytic conductor: $$3.94598$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1950} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1950,\ (\ :1/2),\ -0.581 - 0.813i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.180151944$$ $$L(\frac12)$$ $$\approx$$ $$1.180151944$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - iT$$
5 $$1$$
13 $$1 + (-0.561 - 3.56i)T$$
good7 $$1 - 3.12T + 7T^{2}$$
11 $$1 - 5.12iT - 11T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 - 6iT - 19T^{2}$$
23 $$1 + 5.12iT - 23T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 - 3.12iT - 31T^{2}$$
37 $$1 + 5.12T + 37T^{2}$$
41 $$1 - 0.876iT - 41T^{2}$$
43 $$1 - 6.24iT - 43T^{2}$$
47 $$1 + 6.24T + 47T^{2}$$
53 $$1 + 13.3iT - 53T^{2}$$
59 $$1 - 1.12iT - 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 - 4.87T + 67T^{2}$$
71 $$1 - 10.2iT - 71T^{2}$$
73 $$1 - 13.1T + 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 6.24T + 83T^{2}$$
89 $$1 + 3.12iT - 89T^{2}$$
97 $$1 + 13.1T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$