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 + 4.15·2-s − 4.65·3-s + 9.22·4-s − 19.9·5-s − 19.3·6-s + 27.8·7-s + 5.09·8-s − 5.29·9-s − 82.6·10-s − 0.733·11-s − 42.9·12-s + 67.6·13-s + 115.·14-s + 92.7·15-s − 52.6·16-s − 45.0·17-s − 21.9·18-s + 96.4·19-s − 183.·20-s − 129.·21-s − 3.04·22-s − 76.2·23-s − 23.7·24-s + 271.·25-s + 280.·26-s + 150.·27-s + 257.·28-s + ⋯
 L(s)  = 1 + 1.46·2-s − 0.896·3-s + 1.15·4-s − 1.78·5-s − 1.31·6-s + 1.50·7-s + 0.224·8-s − 0.196·9-s − 2.61·10-s − 0.0200·11-s − 1.03·12-s + 1.44·13-s + 2.20·14-s + 1.59·15-s − 0.823·16-s − 0.642·17-s − 0.287·18-s + 1.16·19-s − 2.05·20-s − 1.34·21-s − 0.0294·22-s − 0.691·23-s − 0.201·24-s + 2.16·25-s + 2.11·26-s + 1.07·27-s + 1.73·28-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 - 4.15T + 8T^{2}$$
3 $$1 + 4.65T + 27T^{2}$$
5 $$1 + 19.9T + 125T^{2}$$
7 $$1 - 27.8T + 343T^{2}$$
11 $$1 + 0.733T + 1.33e3T^{2}$$
13 $$1 - 67.6T + 2.19e3T^{2}$$
17 $$1 + 45.0T + 4.91e3T^{2}$$
19 $$1 - 96.4T + 6.85e3T^{2}$$
23 $$1 + 76.2T + 1.21e4T^{2}$$
29 $$1 + 182.T + 2.43e4T^{2}$$
31 $$1 + 37.9T + 2.97e4T^{2}$$
37 $$1 - 95.8T + 5.06e4T^{2}$$
41 $$1 - 33.5T + 6.89e4T^{2}$$
47 $$1 - 533.T + 1.03e5T^{2}$$
53 $$1 + 0.134T + 1.48e5T^{2}$$
59 $$1 + 110.T + 2.05e5T^{2}$$
61 $$1 - 533.T + 2.26e5T^{2}$$
67 $$1 + 411.T + 3.00e5T^{2}$$
71 $$1 + 216.T + 3.57e5T^{2}$$
73 $$1 + 136.T + 3.89e5T^{2}$$
79 $$1 - 81.5T + 4.93e5T^{2}$$
83 $$1 + 926.T + 5.71e5T^{2}$$
89 $$1 + 1.12e3T + 7.04e5T^{2}$$
97 $$1 + 1.51e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$