# Properties

 Label 2-43e2-1.1-c3-0-366 Degree $2$ Conductor $1849$ Sign $-1$ Analytic cond. $109.094$ Root an. cond. $10.4448$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3.72·2-s − 5.79·3-s + 5.84·4-s + 14.8·5-s − 21.5·6-s + 12.2·7-s − 8.00·8-s + 6.59·9-s + 55.1·10-s + 2.05·11-s − 33.8·12-s + 9.53·13-s + 45.4·14-s − 85.8·15-s − 76.5·16-s − 74.1·17-s + 24.5·18-s + 103.·19-s + 86.6·20-s − 70.7·21-s + 7.63·22-s − 146.·23-s + 46.4·24-s + 94.4·25-s + 35.4·26-s + 118.·27-s + 71.4·28-s + ⋯
 L(s)  = 1 + 1.31·2-s − 1.11·3-s + 0.731·4-s + 1.32·5-s − 1.46·6-s + 0.659·7-s − 0.353·8-s + 0.244·9-s + 1.74·10-s + 0.0562·11-s − 0.815·12-s + 0.203·13-s + 0.867·14-s − 1.47·15-s − 1.19·16-s − 1.05·17-s + 0.321·18-s + 1.24·19-s + 0.968·20-s − 0.735·21-s + 0.0740·22-s − 1.33·23-s + 0.394·24-s + 0.755·25-s + 0.267·26-s + 0.843·27-s + 0.481·28-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: $-1$ Analytic conductor: $$109.094$$ Root analytic conductor: $$10.4448$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes 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 - 3.72T + 8T^{2}$$
3 $$1 + 5.79T + 27T^{2}$$
5 $$1 - 14.8T + 125T^{2}$$
7 $$1 - 12.2T + 343T^{2}$$
11 $$1 - 2.05T + 1.33e3T^{2}$$
13 $$1 - 9.53T + 2.19e3T^{2}$$
17 $$1 + 74.1T + 4.91e3T^{2}$$
19 $$1 - 103.T + 6.85e3T^{2}$$
23 $$1 + 146.T + 1.21e4T^{2}$$
29 $$1 - 102.T + 2.43e4T^{2}$$
31 $$1 + 308.T + 2.97e4T^{2}$$
37 $$1 + 434.T + 5.06e4T^{2}$$
41 $$1 - 183.T + 6.89e4T^{2}$$
47 $$1 - 348.T + 1.03e5T^{2}$$
53 $$1 + 199.T + 1.48e5T^{2}$$
59 $$1 + 156.T + 2.05e5T^{2}$$
61 $$1 - 261.T + 2.26e5T^{2}$$
67 $$1 + 43.3T + 3.00e5T^{2}$$
71 $$1 - 654.T + 3.57e5T^{2}$$
73 $$1 + 868.T + 3.89e5T^{2}$$
79 $$1 + 1.35e3T + 4.93e5T^{2}$$
83 $$1 - 166.T + 5.71e5T^{2}$$
89 $$1 - 1.18e3T + 7.04e5T^{2}$$
97 $$1 + 398.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$