# Properties

 Degree 2 Conductor 43 Sign $-0.0401 - 0.999i$ Motivic weight 10 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.11i·2-s + (−393. + 227. i)3-s + 973.·4-s + (−1.94e3 + 1.12e3i)5-s + (−1.61e3 − 2.79e3i)6-s + (8.90e3 + 5.14e3i)7-s + 1.42e4i·8-s + (7.37e4 − 1.27e5i)9-s + (−7.97e3 − 1.38e4i)10-s + 2.29e5·11-s + (−3.83e5 + 2.21e5i)12-s + (5.23e4 − 9.06e4i)13-s + (−3.65e4 + 6.33e4i)14-s + (5.10e5 − 8.83e5i)15-s + 8.95e5·16-s + (9.78e5 − 1.69e6i)17-s + ⋯
 L(s)  = 1 + 0.222i·2-s + (−1.61 + 0.935i)3-s + 0.950·4-s + (−0.621 + 0.359i)5-s + (−0.207 − 0.359i)6-s + (0.530 + 0.306i)7-s + 0.433i·8-s + (1.24 − 2.16i)9-s + (−0.0797 − 0.138i)10-s + 1.42·11-s + (−1.53 + 0.889i)12-s + (0.140 − 0.244i)13-s + (−0.0680 + 0.117i)14-s + (0.671 − 1.16i)15-s + 0.854·16-s + (0.689 − 1.19i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.0401 - 0.999i$ motivic weight = $$10$$ character : $\chi_{43} (7, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :5),\ -0.0401 - 0.999i)$$ $$L(\frac{11}{2})$$ $$\approx$$ $$1.06127 + 1.10480i$$ $$L(\frac12)$$ $$\approx$$ $$1.06127 + 1.10480i$$ $$L(6)$$ 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 + (-1.46e8 - 6.76e6i)T$$
good2 $$1 - 7.11iT - 1.02e3T^{2}$$
3 $$1 + (393. - 227. i)T + (2.95e4 - 5.11e4i)T^{2}$$
5 $$1 + (1.94e3 - 1.12e3i)T + (4.88e6 - 8.45e6i)T^{2}$$
7 $$1 + (-8.90e3 - 5.14e3i)T + (1.41e8 + 2.44e8i)T^{2}$$
11 $$1 - 2.29e5T + 2.59e10T^{2}$$
13 $$1 + (-5.23e4 + 9.06e4i)T + (-6.89e10 - 1.19e11i)T^{2}$$
17 $$1 + (-9.78e5 + 1.69e6i)T + (-1.00e12 - 1.74e12i)T^{2}$$
19 $$1 + (-9.72e4 + 5.61e4i)T + (3.06e12 - 5.30e12i)T^{2}$$
23 $$1 + (-2.53e6 - 4.38e6i)T + (-2.07e13 + 3.58e13i)T^{2}$$
29 $$1 + (-1.83e6 - 1.05e6i)T + (2.10e14 + 3.64e14i)T^{2}$$
31 $$1 + (-3.80e6 - 6.59e6i)T + (-4.09e14 + 7.09e14i)T^{2}$$
37 $$1 + (-2.72e7 + 1.57e7i)T + (2.40e15 - 4.16e15i)T^{2}$$
41 $$1 - 2.35e7T + 1.34e16T^{2}$$
47 $$1 + 2.97e8T + 5.25e16T^{2}$$
53 $$1 + (-2.77e8 - 4.80e8i)T + (-8.74e16 + 1.51e17i)T^{2}$$
59 $$1 - 1.27e9T + 5.11e17T^{2}$$
61 $$1 + (1.25e9 + 7.22e8i)T + (3.56e17 + 6.17e17i)T^{2}$$
67 $$1 + (6.63e8 + 1.14e9i)T + (-9.11e17 + 1.57e18i)T^{2}$$
71 $$1 + (-2.42e8 - 1.39e8i)T + (1.62e18 + 2.81e18i)T^{2}$$
73 $$1 + (6.34e8 + 3.66e8i)T + (2.14e18 + 3.72e18i)T^{2}$$
79 $$1 + (-1.18e9 + 2.05e9i)T + (-4.73e18 - 8.19e18i)T^{2}$$
83 $$1 + (-3.79e9 - 6.57e9i)T + (-7.75e18 + 1.34e19i)T^{2}$$
89 $$1 + (2.16e9 - 1.25e9i)T + (1.55e19 - 2.70e19i)T^{2}$$
97 $$1 - 1.27e10T + 7.37e19T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}