# Properties

 Degree 2 Conductor 43 Sign $0.586 - 0.809i$ Motivic weight 9 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 26.2·2-s + (−64.5 + 111. i)3-s + 178.·4-s + (−856. + 1.48e3i)5-s + (1.69e3 − 2.93e3i)6-s + (−4.32e3 − 7.49e3i)7-s + 8.76e3·8-s + (1.50e3 + 2.61e3i)9-s + (2.25e4 − 3.89e4i)10-s + 4.66e3·11-s + (−1.15e4 + 1.99e4i)12-s + (−9.88e4 − 1.71e5i)13-s + (1.13e5 + 1.96e5i)14-s + (−1.10e5 − 1.91e5i)15-s − 3.21e5·16-s + (−3.49e4 − 6.05e4i)17-s + ⋯
 L(s)  = 1 − 1.16·2-s + (−0.460 + 0.796i)3-s + 0.348·4-s + (−0.612 + 1.06i)5-s + (0.534 − 0.925i)6-s + (−0.680 − 1.17i)7-s + 0.756·8-s + (0.0766 + 0.132i)9-s + (0.711 − 1.23i)10-s + 0.0960·11-s + (−0.160 + 0.277i)12-s + (−0.959 − 1.66i)13-s + (0.790 + 1.36i)14-s + (−0.564 − 0.976i)15-s − 1.22·16-s + (−0.101 − 0.175i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.586 - 0.809i$ motivic weight = $$9$$ character : $\chi_{43} (6, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :9/2),\ 0.586 - 0.809i)$$ $$L(5)$$ $$\approx$$ $$0.369260 + 0.188522i$$ $$L(\frac12)$$ $$\approx$$ $$0.369260 + 0.188522i$$ $$L(\frac{11}{2})$$ 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 + (-2.58e6 - 2.22e7i)T$$
good2 $$1 + 26.2T + 512T^{2}$$
3 $$1 + (64.5 - 111. i)T + (-9.84e3 - 1.70e4i)T^{2}$$
5 $$1 + (856. - 1.48e3i)T + (-9.76e5 - 1.69e6i)T^{2}$$
7 $$1 + (4.32e3 + 7.49e3i)T + (-2.01e7 + 3.49e7i)T^{2}$$
11 $$1 - 4.66e3T + 2.35e9T^{2}$$
13 $$1 + (9.88e4 + 1.71e5i)T + (-5.30e9 + 9.18e9i)T^{2}$$
17 $$1 + (3.49e4 + 6.05e4i)T + (-5.92e10 + 1.02e11i)T^{2}$$
19 $$1 + (-7.08e4 + 1.22e5i)T + (-1.61e11 - 2.79e11i)T^{2}$$
23 $$1 + (-3.83e5 + 6.64e5i)T + (-9.00e11 - 1.55e12i)T^{2}$$
29 $$1 + (-3.46e6 - 5.99e6i)T + (-7.25e12 + 1.25e13i)T^{2}$$
31 $$1 + (1.22e6 - 2.11e6i)T + (-1.32e13 - 2.28e13i)T^{2}$$
37 $$1 + (-6.26e6 + 1.08e7i)T + (-6.49e13 - 1.12e14i)T^{2}$$
41 $$1 - 1.39e6T + 3.27e14T^{2}$$
47 $$1 + 2.95e7T + 1.11e15T^{2}$$
53 $$1 + (-8.91e5 + 1.54e6i)T + (-1.64e15 - 2.85e15i)T^{2}$$
59 $$1 + 1.36e8T + 8.66e15T^{2}$$
61 $$1 + (7.89e7 + 1.36e8i)T + (-5.84e15 + 1.01e16i)T^{2}$$
67 $$1 + (4.47e7 - 7.75e7i)T + (-1.36e16 - 2.35e16i)T^{2}$$
71 $$1 + (-1.97e7 - 3.41e7i)T + (-2.29e16 + 3.97e16i)T^{2}$$
73 $$1 + (-1.60e8 - 2.77e8i)T + (-2.94e16 + 5.09e16i)T^{2}$$
79 $$1 + (-2.12e8 - 3.68e8i)T + (-5.99e16 + 1.03e17i)T^{2}$$
83 $$1 + (-1.83e8 + 3.17e8i)T + (-9.34e16 - 1.61e17i)T^{2}$$
89 $$1 + (-1.60e8 + 2.77e8i)T + (-1.75e17 - 3.03e17i)T^{2}$$
97 $$1 - 1.21e9T + 7.60e17T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}