# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.44·2-s + 2.44·3-s + 3.99·4-s + 2.44·5-s − 5.99·6-s − 2.44·7-s − 4.89·8-s + 2.99·9-s − 5.99·10-s − 11-s + 9.79·12-s − 3·13-s + 5.99·14-s + 5.99·15-s + 3.99·16-s − 7·17-s − 7.34·18-s − 4.89·19-s + 9.79·20-s − 5.99·21-s + 2.44·22-s + 23-s − 11.9·24-s + 0.999·25-s + 7.34·26-s − 9.79·28-s + 2.44·29-s + ⋯
 L(s)  = 1 − 1.73·2-s + 1.41·3-s + 1.99·4-s + 1.09·5-s − 2.44·6-s − 0.925·7-s − 1.73·8-s + 0.999·9-s − 1.89·10-s − 0.301·11-s + 2.82·12-s − 0.832·13-s + 1.60·14-s + 1.54·15-s + 0.999·16-s − 1.69·17-s − 1.73·18-s − 1.12·19-s + 2.19·20-s − 1.30·21-s + 0.522·22-s + 0.208·23-s − 2.44·24-s + 0.199·25-s + 1.44·26-s − 1.85·28-s + 0.454·29-s + ⋯

## Functional equation

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

## Invariants

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

## 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$$F_p(T)$
bad43 $$1$$
good2 $$1 + 2.44T + 2T^{2}$$
3 $$1 - 2.44T + 3T^{2}$$
5 $$1 - 2.44T + 5T^{2}$$
7 $$1 + 2.44T + 7T^{2}$$
11 $$1 + T + 11T^{2}$$
13 $$1 + 3T + 13T^{2}$$
17 $$1 + 7T + 17T^{2}$$
19 $$1 + 4.89T + 19T^{2}$$
23 $$1 - T + 23T^{2}$$
29 $$1 - 2.44T + 29T^{2}$$
31 $$1 + 3T + 31T^{2}$$
37 $$1 - 4.89T + 37T^{2}$$
41 $$1 + 5T + 41T^{2}$$
47 $$1 + 10T + 47T^{2}$$
53 $$1 + T + 53T^{2}$$
59 $$1 + 10T + 59T^{2}$$
61 $$1 + 7.34T + 61T^{2}$$
67 $$1 - 9T + 67T^{2}$$
71 $$1 + 4.89T + 71T^{2}$$
73 $$1 - 12.2T + 73T^{2}$$
79 $$1 + 6T + 79T^{2}$$
83 $$1 - T + 83T^{2}$$
89 $$1 - 17.1T + 89T^{2}$$
97 $$1 - 11T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$