# Properties

 Label 2-43-1.1-c5-0-4 Degree $2$ Conductor $43$ Sign $-1$ Analytic cond. $6.89650$ Root an. cond. $2.62611$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 11.1·2-s − 28.1·3-s + 92.7·4-s − 10.1·5-s + 314.·6-s + 135.·7-s − 678.·8-s + 550.·9-s + 113.·10-s − 74.1·11-s − 2.61e3·12-s − 252.·13-s − 1.51e3·14-s + 285.·15-s + 4.60e3·16-s + 233.·17-s − 6.15e3·18-s + 550.·19-s − 941.·20-s − 3.82e3·21-s + 827.·22-s + 1.95e3·23-s + 1.91e4·24-s − 3.02e3·25-s + 2.81e3·26-s − 8.67e3·27-s + 1.25e4·28-s + ⋯
 L(s)  = 1 − 1.97·2-s − 1.80·3-s + 2.89·4-s − 0.181·5-s + 3.56·6-s + 1.04·7-s − 3.74·8-s + 2.26·9-s + 0.358·10-s − 0.184·11-s − 5.23·12-s − 0.414·13-s − 2.06·14-s + 0.328·15-s + 4.49·16-s + 0.195·17-s − 4.47·18-s + 0.349·19-s − 0.526·20-s − 1.89·21-s + 0.364·22-s + 0.769·23-s + 6.77·24-s − 0.967·25-s + 0.817·26-s − 2.29·27-s + 3.03·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-1$ Analytic conductor: $$6.89650$$ Root analytic conductor: $$2.62611$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{43} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 43,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + 1.84e3T$$
good2 $$1 + 11.1T + 32T^{2}$$
3 $$1 + 28.1T + 243T^{2}$$
5 $$1 + 10.1T + 3.12e3T^{2}$$
7 $$1 - 135.T + 1.68e4T^{2}$$
11 $$1 + 74.1T + 1.61e5T^{2}$$
13 $$1 + 252.T + 3.71e5T^{2}$$
17 $$1 - 233.T + 1.41e6T^{2}$$
19 $$1 - 550.T + 2.47e6T^{2}$$
23 $$1 - 1.95e3T + 6.43e6T^{2}$$
29 $$1 + 4.67e3T + 2.05e7T^{2}$$
31 $$1 - 3.33e3T + 2.86e7T^{2}$$
37 $$1 + 1.34e4T + 6.93e7T^{2}$$
41 $$1 - 8.20e3T + 1.15e8T^{2}$$
47 $$1 + 2.20e4T + 2.29e8T^{2}$$
53 $$1 - 6.34e3T + 4.18e8T^{2}$$
59 $$1 + 1.49e4T + 7.14e8T^{2}$$
61 $$1 + 2.38e4T + 8.44e8T^{2}$$
67 $$1 - 3.86e4T + 1.35e9T^{2}$$
71 $$1 - 4.91e3T + 1.80e9T^{2}$$
73 $$1 - 5.26e3T + 2.07e9T^{2}$$
79 $$1 + 2.68e4T + 3.07e9T^{2}$$
83 $$1 - 4.84e4T + 3.93e9T^{2}$$
89 $$1 + 1.17e5T + 5.58e9T^{2}$$
97 $$1 + 6.23e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$