# Properties

 Degree 2 Conductor 43 Sign $-0.999 + 0.0292i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.49 − 5.17i)2-s + (−7.43 − 0.557i)3-s + (−10.5 − 13.2i)4-s + (9.68 − 31.4i)5-s + (−21.4 + 37.0i)6-s + (−66.6 + 38.4i)7-s + (−5.54 + 1.26i)8-s + (−25.1 − 3.78i)9-s + (−138. − 128. i)10-s + (42.0 − 52.7i)11-s + (71.3 + 104. i)12-s + (46.0 − 42.7i)13-s + (33.0 + 440. i)14-s + (−89.5 + 228. i)15-s + (53.2 − 233. i)16-s + (364. − 112. i)17-s + ⋯
 L(s)  = 1 + (0.623 − 1.29i)2-s + (−0.826 − 0.0619i)3-s + (−0.662 − 0.830i)4-s + (0.387 − 1.25i)5-s + (−0.594 + 1.03i)6-s + (−1.36 + 0.785i)7-s + (−0.0866 + 0.0197i)8-s + (−0.309 − 0.0467i)9-s + (−1.38 − 1.28i)10-s + (0.347 − 0.435i)11-s + (0.495 + 0.726i)12-s + (0.272 − 0.252i)13-s + (0.168 + 2.24i)14-s + (−0.397 + 1.01i)15-s + (0.207 − 0.910i)16-s + (1.25 − 0.388i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.999 + 0.0292i$ motivic weight = $$4$$ character : $\chi_{43} (3, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ -0.999 + 0.0292i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.0190019 - 1.29799i$$ $$L(\frac12)$$ $$\approx$$ $$0.0190019 - 1.29799i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 $$1 + (-538. - 1.76e3i)T$$
good2 $$1 + (-2.49 + 5.17i)T + (-9.97 - 12.5i)T^{2}$$
3 $$1 + (7.43 + 0.557i)T + (80.0 + 12.0i)T^{2}$$
5 $$1 + (-9.68 + 31.4i)T + (-516. - 352. i)T^{2}$$
7 $$1 + (66.6 - 38.4i)T + (1.20e3 - 2.07e3i)T^{2}$$
11 $$1 + (-42.0 + 52.7i)T + (-3.25e3 - 1.42e4i)T^{2}$$
13 $$1 + (-46.0 + 42.7i)T + (2.13e3 - 2.84e4i)T^{2}$$
17 $$1 + (-364. + 112. i)T + (6.90e4 - 4.70e4i)T^{2}$$
19 $$1 + (36.5 + 242. i)T + (-1.24e5 + 3.84e4i)T^{2}$$
23 $$1 + (190. + 485. i)T + (-2.05e5 + 1.90e5i)T^{2}$$
29 $$1 + (-1.04e3 + 77.9i)T + (6.99e5 - 1.05e5i)T^{2}$$
31 $$1 + (1.41e3 - 962. i)T + (3.37e5 - 8.59e5i)T^{2}$$
37 $$1 + (-814. - 470. i)T + (9.37e5 + 1.62e6i)T^{2}$$
41 $$1 + (-844. - 406. i)T + (1.76e6 + 2.20e6i)T^{2}$$
47 $$1 + (217. + 273. i)T + (-1.08e6 + 4.75e6i)T^{2}$$
53 $$1 + (2.55e3 + 2.36e3i)T + (5.89e5 + 7.86e6i)T^{2}$$
59 $$1 + (319. - 1.40e3i)T + (-1.09e7 - 5.25e6i)T^{2}$$
61 $$1 + (1.74e3 - 2.56e3i)T + (-5.05e6 - 1.28e7i)T^{2}$$
67 $$1 + (-8.01e3 + 1.20e3i)T + (1.92e7 - 5.93e6i)T^{2}$$
71 $$1 + (-676. - 265. i)T + (1.86e7 + 1.72e7i)T^{2}$$
73 $$1 + (4.05e3 + 4.37e3i)T + (-2.12e6 + 2.83e7i)T^{2}$$
79 $$1 + (-3.18e3 - 5.51e3i)T + (-1.94e7 + 3.37e7i)T^{2}$$
83 $$1 + (-295. + 3.93e3i)T + (-4.69e7 - 7.07e6i)T^{2}$$
89 $$1 + (6.41e3 + 480. i)T + (6.20e7 + 9.35e6i)T^{2}$$
97 $$1 + (-2.03e3 + 2.55e3i)T + (-1.96e7 - 8.63e7i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}