# Properties

 Label 2-5733-1.1-c1-0-139 Degree $2$ Conductor $5733$ Sign $-1$ Analytic cond. $45.7782$ Root an. cond. $6.76596$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.470·2-s − 1.77·4-s + 0.529·5-s + 1.77·8-s − 0.249·10-s + 2.24·11-s − 13-s + 2.71·16-s − 1.30·17-s + 1.47·19-s − 0.941·20-s − 1.05·22-s − 5.83·23-s − 4.71·25-s + 0.470·26-s − 5.22·29-s + 7.02·31-s − 4.83·32-s + 0.615·34-s − 2.36·37-s − 0.692·38-s + 0.941·40-s + 6.49·41-s + 11.3·43-s − 4.00·44-s + 2.74·46-s + 8.58·47-s + ⋯
 L(s)  = 1 − 0.332·2-s − 0.889·4-s + 0.236·5-s + 0.628·8-s − 0.0787·10-s + 0.678·11-s − 0.277·13-s + 0.679·16-s − 0.317·17-s + 0.337·19-s − 0.210·20-s − 0.225·22-s − 1.21·23-s − 0.943·25-s + 0.0923·26-s − 0.969·29-s + 1.26·31-s − 0.855·32-s + 0.105·34-s − 0.389·37-s − 0.112·38-s + 0.148·40-s + 1.01·41-s + 1.73·43-s − 0.603·44-s + 0.405·46-s + 1.25·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5733$$    =    $$3^{2} \cdot 7^{2} \cdot 13$$ Sign: $-1$ Analytic conductor: $$45.7782$$ Root analytic conductor: $$6.76596$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5733} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 5733,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
13 $$1 + T$$
good2 $$1 + 0.470T + 2T^{2}$$
5 $$1 - 0.529T + 5T^{2}$$
11 $$1 - 2.24T + 11T^{2}$$
17 $$1 + 1.30T + 17T^{2}$$
19 $$1 - 1.47T + 19T^{2}$$
23 $$1 + 5.83T + 23T^{2}$$
29 $$1 + 5.22T + 29T^{2}$$
31 $$1 - 7.02T + 31T^{2}$$
37 $$1 + 2.36T + 37T^{2}$$
41 $$1 - 6.49T + 41T^{2}$$
43 $$1 - 11.3T + 43T^{2}$$
47 $$1 - 8.58T + 47T^{2}$$
53 $$1 + 11.2T + 53T^{2}$$
59 $$1 + 12.1T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + 15.9T + 67T^{2}$$
71 $$1 + 1.19T + 71T^{2}$$
73 $$1 + 7.64T + 73T^{2}$$
79 $$1 + 1.33T + 79T^{2}$$
83 $$1 + 16.3T + 83T^{2}$$
89 $$1 - 6.91T + 89T^{2}$$
97 $$1 - 3.47T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$