# Properties

 Label 2-43-1.1-c7-0-20 Degree $2$ Conductor $43$ Sign $-1$ Analytic cond. $13.4325$ Root an. cond. $3.66504$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 12.3·2-s − 46.7·3-s + 23.7·4-s + 330.·5-s − 575.·6-s − 467.·7-s − 1.28e3·8-s − 4.26·9-s + 4.06e3·10-s − 359.·11-s − 1.10e3·12-s − 1.14e4·13-s − 5.75e3·14-s − 1.54e4·15-s − 1.88e4·16-s − 1.90e4·17-s − 52.5·18-s − 3.67e3·19-s + 7.83e3·20-s + 2.18e4·21-s − 4.42e3·22-s − 3.78e3·23-s + 6.00e4·24-s + 3.08e4·25-s − 1.40e5·26-s + 1.02e5·27-s − 1.10e4·28-s + ⋯
 L(s)  = 1 + 1.08·2-s − 0.999·3-s + 0.185·4-s + 1.18·5-s − 1.08·6-s − 0.515·7-s − 0.886·8-s − 0.00195·9-s + 1.28·10-s − 0.0813·11-s − 0.185·12-s − 1.44·13-s − 0.560·14-s − 1.18·15-s − 1.15·16-s − 0.937·17-s − 0.00212·18-s − 0.122·19-s + 0.219·20-s + 0.514·21-s − 0.0886·22-s − 0.0648·23-s + 0.886·24-s + 0.395·25-s − 1.57·26-s + 1.00·27-s − 0.0955·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-1$ Analytic conductor: $$13.4325$$ Root analytic conductor: $$3.66504$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{43} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 43,\ (\ :7/2),\ -1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 - 7.95e4T$$
good2 $$1 - 12.3T + 128T^{2}$$
3 $$1 + 46.7T + 2.18e3T^{2}$$
5 $$1 - 330.T + 7.81e4T^{2}$$
7 $$1 + 467.T + 8.23e5T^{2}$$
11 $$1 + 359.T + 1.94e7T^{2}$$
13 $$1 + 1.14e4T + 6.27e7T^{2}$$
17 $$1 + 1.90e4T + 4.10e8T^{2}$$
19 $$1 + 3.67e3T + 8.93e8T^{2}$$
23 $$1 + 3.78e3T + 3.40e9T^{2}$$
29 $$1 - 1.15e5T + 1.72e10T^{2}$$
31 $$1 - 1.51e4T + 2.75e10T^{2}$$
37 $$1 + 3.57e5T + 9.49e10T^{2}$$
41 $$1 - 5.40e5T + 1.94e11T^{2}$$
47 $$1 - 2.89e5T + 5.06e11T^{2}$$
53 $$1 + 3.29e5T + 1.17e12T^{2}$$
59 $$1 - 2.44e6T + 2.48e12T^{2}$$
61 $$1 + 1.25e6T + 3.14e12T^{2}$$
67 $$1 + 1.70e6T + 6.06e12T^{2}$$
71 $$1 + 3.75e6T + 9.09e12T^{2}$$
73 $$1 + 5.83e6T + 1.10e13T^{2}$$
79 $$1 - 3.48e6T + 1.92e13T^{2}$$
83 $$1 - 5.17e6T + 2.71e13T^{2}$$
89 $$1 - 5.98e6T + 4.42e13T^{2}$$
97 $$1 + 5.18e6T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$