# Properties

 Label 2-43-43.6-c5-0-15 Degree $2$ Conductor $43$ Sign $-0.880 + 0.473i$ Analytic cond. $6.89650$ Root an. cond. $2.62611$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.74·2-s + (7.37 − 12.7i)3-s − 28.9·4-s + (−8.29 + 14.3i)5-s + (12.8 − 22.2i)6-s + (−98.7 − 171. i)7-s − 106.·8-s + (12.6 + 21.9i)9-s + (−14.4 + 25.0i)10-s − 376.·11-s + (−213. + 369. i)12-s + (−55.5 − 96.2i)13-s + (−172. − 298. i)14-s + (122. + 212. i)15-s + 741.·16-s + (−786. − 1.36e3i)17-s + ⋯
 L(s)  = 1 + 0.308·2-s + (0.473 − 0.819i)3-s − 0.904·4-s + (−0.148 + 0.257i)5-s + (0.145 − 0.252i)6-s + (−0.761 − 1.31i)7-s − 0.587·8-s + (0.0522 + 0.0905i)9-s + (−0.0457 + 0.0792i)10-s − 0.938·11-s + (−0.428 + 0.741i)12-s + (−0.0911 − 0.157i)13-s + (−0.234 − 0.406i)14-s + (0.140 + 0.243i)15-s + 0.724·16-s + (−0.659 − 1.14i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-0.880 + 0.473i$ Analytic conductor: $$6.89650$$ Root analytic conductor: $$2.62611$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{43} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :5/2),\ -0.880 + 0.473i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.232480 - 0.923063i$$ $$L(\frac12)$$ $$\approx$$ $$0.232480 - 0.923063i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-3.98e3 + 1.14e4i)T$$
good2 $$1 - 1.74T + 32T^{2}$$
3 $$1 + (-7.37 + 12.7i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (8.29 - 14.3i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (98.7 + 171. i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + 376.T + 1.61e5T^{2}$$
13 $$1 + (55.5 + 96.2i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + (786. + 1.36e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (-444. + 769. i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-428. + 741. i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (-12.2 - 21.1i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-2.22e3 + 3.84e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (3.15e3 - 5.46e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 - 5.20e3T + 1.15e8T^{2}$$
47 $$1 + 1.33e4T + 2.29e8T^{2}$$
53 $$1 + (-2.12e3 + 3.68e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + 4.94e4T + 7.14e8T^{2}$$
61 $$1 + (1.70e4 + 2.94e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-2.10e4 + 3.64e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (1.94e4 + 3.36e4i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (1.27e4 + 2.20e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (-4.78e4 - 8.29e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (1.03e3 - 1.79e3i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + (7.30e4 - 1.26e5i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 1.89e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$