# Properties

 Label 2-4650-5.4-c1-0-59 Degree $2$ Conductor $4650$ Sign $0.447 - 0.894i$ Analytic cond. $37.1304$ Root an. cond. $6.09347$ 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 + 2.56i·7-s − i·8-s − 9-s + 2.56·11-s − i·12-s − 2i·13-s − 2.56·14-s + 16-s − 3.12i·17-s − i·18-s + 7.68·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.968i·7-s − 0.353i·8-s − 0.333·9-s + 0.772·11-s − 0.288i·12-s − 0.554i·13-s − 0.684·14-s + 0.250·16-s − 0.757i·17-s − 0.235i·18-s + 1.76·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{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: $$4650$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 31$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$37.1304$$ Root analytic conductor: $$6.09347$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4650} (3349, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4650,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.911338867$$ $$L(\frac12)$$ $$\approx$$ $$1.911338867$$ $$L(\frac{3}{2})$$ 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$$
31 $$1 - T$$
good7 $$1 - 2.56iT - 7T^{2}$$
11 $$1 - 2.56T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 + 3.12iT - 17T^{2}$$
19 $$1 - 7.68T + 19T^{2}$$
23 $$1 + 1.43iT - 23T^{2}$$
29 $$1 + 7.12T + 29T^{2}$$
37 $$1 + 3.12iT - 37T^{2}$$
41 $$1 - 7.12T + 41T^{2}$$
43 $$1 + 12.8iT - 43T^{2}$$
47 $$1 - 5.12iT - 47T^{2}$$
53 $$1 + 7.43iT - 53T^{2}$$
59 $$1 - 13.1T + 59T^{2}$$
61 $$1 - 6T + 61T^{2}$$
67 $$1 + 15.3iT - 67T^{2}$$
71 $$1 + 7.68T + 71T^{2}$$
73 $$1 - 10.8iT - 73T^{2}$$
79 $$1 - 4.31T + 79T^{2}$$
83 $$1 + 14.2iT - 83T^{2}$$
89 $$1 - 13.6T + 89T^{2}$$
97 $$1 + 6iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$