# Properties

 Degree $2$ Conductor $1849$ Sign $-1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.05·2-s − 9.36·3-s + 17.5·4-s − 14.1·5-s − 47.3·6-s + 13.7·7-s + 48.1·8-s + 60.6·9-s − 71.4·10-s + 10.3·11-s − 164.·12-s − 61.4·13-s + 69.2·14-s + 132.·15-s + 103.·16-s − 24.4·17-s + 306.·18-s + 65.5·19-s − 247.·20-s − 128.·21-s + 52.3·22-s + 23.4·23-s − 450.·24-s + 75.0·25-s − 310.·26-s − 315.·27-s + 240.·28-s + ⋯
 L(s)  = 1 + 1.78·2-s − 1.80·3-s + 2.19·4-s − 1.26·5-s − 3.21·6-s + 0.740·7-s + 2.12·8-s + 2.24·9-s − 2.25·10-s + 0.284·11-s − 3.94·12-s − 1.31·13-s + 1.32·14-s + 2.27·15-s + 1.60·16-s − 0.348·17-s + 4.01·18-s + 0.791·19-s − 2.77·20-s − 1.33·21-s + 0.507·22-s + 0.212·23-s − 3.83·24-s + 0.600·25-s − 2.34·26-s − 2.24·27-s + 1.62·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1849$$    =    $$43^{2}$$ Sign: $-1$ Motivic weight: $$3$$ Character: $\chi_{1849} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1849,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1$$
good2 $$1 - 5.05T + 8T^{2}$$
3 $$1 + 9.36T + 27T^{2}$$
5 $$1 + 14.1T + 125T^{2}$$
7 $$1 - 13.7T + 343T^{2}$$
11 $$1 - 10.3T + 1.33e3T^{2}$$
13 $$1 + 61.4T + 2.19e3T^{2}$$
17 $$1 + 24.4T + 4.91e3T^{2}$$
19 $$1 - 65.5T + 6.85e3T^{2}$$
23 $$1 - 23.4T + 1.21e4T^{2}$$
29 $$1 - 236.T + 2.43e4T^{2}$$
31 $$1 + 210.T + 2.97e4T^{2}$$
37 $$1 - 316.T + 5.06e4T^{2}$$
41 $$1 - 13.2T + 6.89e4T^{2}$$
47 $$1 - 450.T + 1.03e5T^{2}$$
53 $$1 + 205.T + 1.48e5T^{2}$$
59 $$1 - 334.T + 2.05e5T^{2}$$
61 $$1 + 811.T + 2.26e5T^{2}$$
67 $$1 + 113.T + 3.00e5T^{2}$$
71 $$1 - 53.8T + 3.57e5T^{2}$$
73 $$1 - 108.T + 3.89e5T^{2}$$
79 $$1 + 723.T + 4.93e5T^{2}$$
83 $$1 - 191.T + 5.71e5T^{2}$$
89 $$1 + 419.T + 7.04e5T^{2}$$
97 $$1 + 953.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$