# Properties

 Degree 2 Conductor 43 Sign $0.755 - 0.655i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.51i·2-s + (20.0 + 11.5i)3-s + 61.6·4-s + (211. + 122. i)5-s + (17.5 − 30.4i)6-s + (−447. + 258. i)7-s − 190. i·8-s + (−96.7 − 167. i)9-s + (185. − 320. i)10-s − 593.·11-s + (1.23e3 + 713. i)12-s + (651. + 1.12e3i)13-s + (391. + 678. i)14-s + (2.82e3 + 4.89e3i)15-s + 3.65e3·16-s + (−1.53e3 − 2.66e3i)17-s + ⋯
 L(s)  = 1 − 0.189i·2-s + (0.742 + 0.428i)3-s + 0.964·4-s + (1.69 + 0.976i)5-s + (0.0812 − 0.140i)6-s + (−1.30 + 0.753i)7-s − 0.372i·8-s + (−0.132 − 0.229i)9-s + (0.185 − 0.320i)10-s − 0.446·11-s + (0.715 + 0.413i)12-s + (0.296 + 0.513i)13-s + (0.142 + 0.247i)14-s + (0.836 + 1.44i)15-s + 0.893·16-s + (−0.313 − 0.542i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.755 - 0.655i$ motivic weight = $$6$$ character : $\chi_{43} (37, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ 0.755 - 0.655i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$2.75033 + 1.02690i$$ $$L(\frac12)$$ $$\approx$$ $$2.75033 + 1.02690i$$ $$L(4)$$ 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 + (5.70e4 + 5.53e4i)T$$
good2 $$1 + 1.51iT - 64T^{2}$$
3 $$1 + (-20.0 - 11.5i)T + (364.5 + 631. i)T^{2}$$
5 $$1 + (-211. - 122. i)T + (7.81e3 + 1.35e4i)T^{2}$$
7 $$1 + (447. - 258. i)T + (5.88e4 - 1.01e5i)T^{2}$$
11 $$1 + 593.T + 1.77e6T^{2}$$
13 $$1 + (-651. - 1.12e3i)T + (-2.41e6 + 4.18e6i)T^{2}$$
17 $$1 + (1.53e3 + 2.66e3i)T + (-1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (3.28e3 + 1.89e3i)T + (2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (-8.01e3 + 1.38e4i)T + (-7.40e7 - 1.28e8i)T^{2}$$
29 $$1 + (-4.72e3 + 2.72e3i)T + (2.97e8 - 5.15e8i)T^{2}$$
31 $$1 + (-7.58e3 + 1.31e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (-2.86e3 - 1.65e3i)T + (1.28e9 + 2.22e9i)T^{2}$$
41 $$1 - 8.53e4T + 4.75e9T^{2}$$
47 $$1 + 2.25e4T + 1.07e10T^{2}$$
53 $$1 + (1.28e5 - 2.23e5i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 + 1.07e5T + 4.21e10T^{2}$$
61 $$1 + (1.91e4 - 1.10e4i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (-2.30e5 + 3.99e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + (-3.46e5 + 2.00e5i)T + (6.40e10 - 1.10e11i)T^{2}$$
73 $$1 + (4.67e5 - 2.70e5i)T + (7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (2.44e5 + 4.22e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + (3.60e5 - 6.23e5i)T + (-1.63e11 - 2.83e11i)T^{2}$$
89 $$1 + (6.12e4 + 3.53e4i)T + (2.48e11 + 4.30e11i)T^{2}$$
97 $$1 + 1.36e6T + 8.32e11T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}