# Properties

 Label 2-43-43.7-c6-0-3 Degree $2$ Conductor $43$ Sign $0.814 - 0.580i$ Analytic cond. $9.89232$ Root an. cond. $3.14520$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 12.4i·2-s + (3.88 − 2.24i)3-s − 91.6·4-s + (−151. + 87.2i)5-s + (−27.9 − 48.4i)6-s + (378. + 218. i)7-s + 344. i·8-s + (−354. + 613. i)9-s + (1.08e3 + 1.88e3i)10-s − 420.·11-s + (−355. + 205. i)12-s + (−379. + 657. i)13-s + (2.72e3 − 4.72e3i)14-s + (−391. + 677. i)15-s − 1.56e3·16-s + (−92.2 + 159. i)17-s + ⋯
 L(s)  = 1 − 1.55i·2-s + (0.143 − 0.0830i)3-s − 1.43·4-s + (−1.20 + 0.697i)5-s + (−0.129 − 0.224i)6-s + (1.10 + 0.637i)7-s + 0.673i·8-s + (−0.486 + 0.842i)9-s + (1.08 + 1.88i)10-s − 0.315·11-s + (−0.205 + 0.118i)12-s + (−0.172 + 0.299i)13-s + (0.993 − 1.72i)14-s + (−0.115 + 0.200i)15-s − 0.382·16-s + (−0.0187 + 0.0325i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.814 - 0.580i$ Analytic conductor: $$9.89232$$ Root analytic conductor: $$3.14520$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{43} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :3),\ 0.814 - 0.580i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.713119 + 0.228332i$$ $$L(\frac12)$$ $$\approx$$ $$0.713119 + 0.228332i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-4.04e4 - 6.84e4i)T$$
good2 $$1 + 12.4iT - 64T^{2}$$
3 $$1 + (-3.88 + 2.24i)T + (364.5 - 631. i)T^{2}$$
5 $$1 + (151. - 87.2i)T + (7.81e3 - 1.35e4i)T^{2}$$
7 $$1 + (-378. - 218. i)T + (5.88e4 + 1.01e5i)T^{2}$$
11 $$1 + 420.T + 1.77e6T^{2}$$
13 $$1 + (379. - 657. i)T + (-2.41e6 - 4.18e6i)T^{2}$$
17 $$1 + (92.2 - 159. i)T + (-1.20e7 - 2.09e7i)T^{2}$$
19 $$1 + (703. - 405. i)T + (2.35e7 - 4.07e7i)T^{2}$$
23 $$1 + (-6.48e3 - 1.12e4i)T + (-7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + (3.08e4 + 1.78e4i)T + (2.97e8 + 5.15e8i)T^{2}$$
31 $$1 + (-1.35e4 - 2.35e4i)T + (-4.43e8 + 7.68e8i)T^{2}$$
37 $$1 + (6.04e4 - 3.48e4i)T + (1.28e9 - 2.22e9i)T^{2}$$
41 $$1 + 1.11e5T + 4.75e9T^{2}$$
47 $$1 - 1.35e5T + 1.07e10T^{2}$$
53 $$1 + (7.54e4 + 1.30e5i)T + (-1.10e10 + 1.91e10i)T^{2}$$
59 $$1 + 6.12e4T + 4.21e10T^{2}$$
61 $$1 + (-7.66e4 - 4.42e4i)T + (2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (-1.17e5 - 2.03e5i)T + (-4.52e10 + 7.83e10i)T^{2}$$
71 $$1 + (1.05e5 + 6.11e4i)T + (6.40e10 + 1.10e11i)T^{2}$$
73 $$1 + (2.97e5 + 1.71e5i)T + (7.56e10 + 1.31e11i)T^{2}$$
79 $$1 + (-4.35e5 + 7.54e5i)T + (-1.21e11 - 2.10e11i)T^{2}$$
83 $$1 + (4.20e5 + 7.28e5i)T + (-1.63e11 + 2.83e11i)T^{2}$$
89 $$1 + (-6.72e5 + 3.88e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 - 1.04e6T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$