# Properties

 Label 2-43-43.6-c5-0-16 Degree $2$ Conductor $43$ Sign $-0.0485 + 0.998i$ 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 + 4.50·2-s + (14.9 − 25.9i)3-s − 11.7·4-s + (16.1 − 28.0i)5-s + (67.4 − 116. i)6-s + (70.7 + 122. i)7-s − 196.·8-s + (−328. − 568. i)9-s + (72.9 − 126. i)10-s + 210.·11-s + (−175. + 304. i)12-s + (55.4 + 95.9i)13-s + (318. + 551. i)14-s + (−485. − 841. i)15-s − 510.·16-s + (856. + 1.48e3i)17-s + ⋯
 L(s)  = 1 + 0.795·2-s + (0.961 − 1.66i)3-s − 0.366·4-s + (0.289 − 0.501i)5-s + (0.765 − 1.32i)6-s + (0.545 + 0.945i)7-s − 1.08·8-s + (−1.35 − 2.33i)9-s + (0.230 − 0.399i)10-s + 0.523·11-s + (−0.352 + 0.610i)12-s + (0.0909 + 0.157i)13-s + (0.434 + 0.752i)14-s + (−0.557 − 0.965i)15-s − 0.498·16-s + (0.719 + 1.24i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-0.0485 + 0.998i$ 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: $$(2,\ 43,\ (\ :5/2),\ -0.0485 + 0.998i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.98666 - 2.08567i$$ $$L(\frac12)$$ $$\approx$$ $$1.98666 - 2.08567i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (7.75e3 + 9.32e3i)T$$
good2 $$1 - 4.50T + 32T^{2}$$
3 $$1 + (-14.9 + 25.9i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (-16.1 + 28.0i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (-70.7 - 122. i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 - 210.T + 1.61e5T^{2}$$
13 $$1 + (-55.4 - 95.9i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + (-856. - 1.48e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (-1.23e3 + 2.13e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-1.24e3 + 2.15e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (-2.34e3 - 4.05e3i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (3.51e3 - 6.08e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (3.16e3 - 5.47e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 + 6.71e3T + 1.15e8T^{2}$$
47 $$1 + 387.T + 2.29e8T^{2}$$
53 $$1 + (2.28e3 - 3.96e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + 4.21e3T + 7.14e8T^{2}$$
61 $$1 + (-7.57e3 - 1.31e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (1.09e4 - 1.89e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (-1.79e4 - 3.10e4i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (3.39e4 + 5.88e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (2.76e4 + 4.78e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (4.75e4 - 8.24e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + (-1.26e4 + 2.19e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 4.61e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$