# Properties

 Degree $2$ Conductor $43$ Sign $0.637 + 0.770i$ Motivic weight $7$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.72 + 7.18i)2-s + (9.19 − 1.38i)3-s + (9.70 + 42.5i)4-s + (445. − 303. i)5-s + (−42.7 + 74.0i)6-s + (−356. − 617. i)7-s + (−1.42e3 − 684. i)8-s + (−2.00e3 + 619. i)9-s + (−370. + 4.94e3i)10-s + (644. − 2.82e3i)11-s + (148. + 377. i)12-s + (−615. − 8.21e3i)13-s + (6.47e3 + 975. i)14-s + (3.68e3 − 3.41e3i)15-s + (8.01e3 − 3.86e3i)16-s + (−2.53e4 − 1.72e4i)17-s + ⋯
 L(s)  = 1 + (−0.506 + 0.634i)2-s + (0.196 − 0.0296i)3-s + (0.0758 + 0.332i)4-s + (1.59 − 1.08i)5-s + (−0.0807 + 0.139i)6-s + (−0.392 − 0.680i)7-s + (−0.980 − 0.472i)8-s + (−0.917 + 0.283i)9-s + (−0.117 + 1.56i)10-s + (0.145 − 0.639i)11-s + (0.0247 + 0.0631i)12-s + (−0.0777 − 1.03i)13-s + (0.630 + 0.0950i)14-s + (0.281 − 0.261i)15-s + (0.489 − 0.235i)16-s + (−1.24 − 0.851i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.637 + 0.770i$ Motivic weight: $$7$$ Character: $\chi_{43} (24, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :7/2),\ 0.637 + 0.770i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$1.29311 - 0.608214i$$ $$L(\frac12)$$ $$\approx$$ $$1.29311 - 0.608214i$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-4.23e5 + 3.03e5i)T$$
good2 $$1 + (5.72 - 7.18i)T + (-28.4 - 124. i)T^{2}$$
3 $$1 + (-9.19 + 1.38i)T + (2.08e3 - 644. i)T^{2}$$
5 $$1 + (-445. + 303. i)T + (2.85e4 - 7.27e4i)T^{2}$$
7 $$1 + (356. + 617. i)T + (-4.11e5 + 7.13e5i)T^{2}$$
11 $$1 + (-644. + 2.82e3i)T + (-1.75e7 - 8.45e6i)T^{2}$$
13 $$1 + (615. + 8.21e3i)T + (-6.20e7 + 9.35e6i)T^{2}$$
17 $$1 + (2.53e4 + 1.72e4i)T + (1.49e8 + 3.81e8i)T^{2}$$
19 $$1 + (-4.50e4 - 1.39e4i)T + (7.38e8 + 5.03e8i)T^{2}$$
23 $$1 + (3.65e4 + 3.39e4i)T + (2.54e8 + 3.39e9i)T^{2}$$
29 $$1 + (-1.54e5 - 2.33e4i)T + (1.64e10 + 5.08e9i)T^{2}$$
31 $$1 + (-4.96e4 - 1.26e5i)T + (-2.01e10 + 1.87e10i)T^{2}$$
37 $$1 + (1.57e5 - 2.72e5i)T + (-4.74e10 - 8.22e10i)T^{2}$$
41 $$1 + (1.78e5 - 2.23e5i)T + (-4.33e10 - 1.89e11i)T^{2}$$
47 $$1 + (8.14e4 + 3.56e5i)T + (-4.56e11 + 2.19e11i)T^{2}$$
53 $$1 + (8.14e3 - 1.08e5i)T + (-1.16e12 - 1.75e11i)T^{2}$$
59 $$1 + (1.68e5 - 8.10e4i)T + (1.55e12 - 1.94e12i)T^{2}$$
61 $$1 + (2.90e4 - 7.41e4i)T + (-2.30e12 - 2.13e12i)T^{2}$$
67 $$1 + (2.08e6 + 6.43e5i)T + (5.00e12 + 3.41e12i)T^{2}$$
71 $$1 + (-2.30e6 + 2.13e6i)T + (6.79e11 - 9.06e12i)T^{2}$$
73 $$1 + (-3.91e5 - 5.23e6i)T + (-1.09e13 + 1.64e12i)T^{2}$$
79 $$1 + (7.28e5 + 1.26e6i)T + (-9.60e12 + 1.66e13i)T^{2}$$
83 $$1 + (-3.54e6 + 5.33e5i)T + (2.59e13 - 7.99e12i)T^{2}$$
89 $$1 + (-3.86e6 + 5.82e5i)T + (4.22e13 - 1.30e13i)T^{2}$$
97 $$1 + (3.21e6 - 1.40e7i)T + (-7.27e13 - 3.50e13i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$