# Properties

 Degree 2 Conductor 43 Sign $-0.739 - 0.673i$ Motivic weight 7 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−9.86 − 12.3i)2-s + (−10.7 − 1.61i)3-s + (−27.2 + 119. i)4-s + (180. + 122. i)5-s + (85.6 + 148. i)6-s + (797. − 1.38e3i)7-s + (−79.5 + 38.2i)8-s + (−1.97e3 − 610. i)9-s + (−258. − 3.44e3i)10-s + (−774. − 3.39e3i)11-s + (484. − 1.23e3i)12-s + (−123. + 1.65e3i)13-s + (−2.49e4 + 3.76e3i)14-s + (−1.73e3 − 1.60e3i)15-s + (1.53e4 + 7.40e3i)16-s + (1.62e4 − 1.10e4i)17-s + ⋯
 L(s)  = 1 + (−0.872 − 1.09i)2-s + (−0.228 − 0.0345i)3-s + (−0.212 + 0.932i)4-s + (0.645 + 0.439i)5-s + (0.161 + 0.280i)6-s + (0.879 − 1.52i)7-s + (−0.0549 + 0.0264i)8-s + (−0.904 − 0.278i)9-s + (−0.0816 − 1.08i)10-s + (−0.175 − 0.768i)11-s + (0.0809 − 0.206i)12-s + (−0.0156 + 0.208i)13-s + (−2.43 + 0.366i)14-s + (−0.132 − 0.123i)15-s + (0.938 + 0.451i)16-s + (0.802 − 0.546i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.739 - 0.673i$ motivic weight = $$7$$ character : $\chi_{43} (9, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :7/2),\ -0.739 - 0.673i)$$ $$L(4)$$ $$\approx$$ $$0.196655 + 0.507886i$$ $$L(\frac12)$$ $$\approx$$ $$0.196655 + 0.507886i$$ $$L(\frac{9}{2})$$ 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 + (-3.23e5 + 4.08e5i)T$$
good2 $$1 + (9.86 + 12.3i)T + (-28.4 + 124. i)T^{2}$$
3 $$1 + (10.7 + 1.61i)T + (2.08e3 + 644. i)T^{2}$$
5 $$1 + (-180. - 122. i)T + (2.85e4 + 7.27e4i)T^{2}$$
7 $$1 + (-797. + 1.38e3i)T + (-4.11e5 - 7.13e5i)T^{2}$$
11 $$1 + (774. + 3.39e3i)T + (-1.75e7 + 8.45e6i)T^{2}$$
13 $$1 + (123. - 1.65e3i)T + (-6.20e7 - 9.35e6i)T^{2}$$
17 $$1 + (-1.62e4 + 1.10e4i)T + (1.49e8 - 3.81e8i)T^{2}$$
19 $$1 + (2.21e4 - 6.81e3i)T + (7.38e8 - 5.03e8i)T^{2}$$
23 $$1 + (5.43e4 - 5.04e4i)T + (2.54e8 - 3.39e9i)T^{2}$$
29 $$1 + (1.92e5 - 2.89e4i)T + (1.64e10 - 5.08e9i)T^{2}$$
31 $$1 + (9.02e4 - 2.30e5i)T + (-2.01e10 - 1.87e10i)T^{2}$$
37 $$1 + (1.68e5 + 2.91e5i)T + (-4.74e10 + 8.22e10i)T^{2}$$
41 $$1 + (2.07e5 + 2.60e5i)T + (-4.33e10 + 1.89e11i)T^{2}$$
47 $$1 + (1.05e5 - 4.63e5i)T + (-4.56e11 - 2.19e11i)T^{2}$$
53 $$1 + (-1.38e5 - 1.85e6i)T + (-1.16e12 + 1.75e11i)T^{2}$$
59 $$1 + (-4.11e5 - 1.98e5i)T + (1.55e12 + 1.94e12i)T^{2}$$
61 $$1 + (4.54e5 + 1.15e6i)T + (-2.30e12 + 2.13e12i)T^{2}$$
67 $$1 + (2.01e6 - 6.22e5i)T + (5.00e12 - 3.41e12i)T^{2}$$
71 $$1 + (1.14e6 + 1.06e6i)T + (6.79e11 + 9.06e12i)T^{2}$$
73 $$1 + (-4.23e5 + 5.64e6i)T + (-1.09e13 - 1.64e12i)T^{2}$$
79 $$1 + (-2.60e6 + 4.51e6i)T + (-9.60e12 - 1.66e13i)T^{2}$$
83 $$1 + (4.87e6 + 7.35e5i)T + (2.59e13 + 7.99e12i)T^{2}$$
89 $$1 + (-1.08e7 - 1.63e6i)T + (4.22e13 + 1.30e13i)T^{2}$$
97 $$1 + (-2.71e5 - 1.18e6i)T + (-7.27e13 + 3.50e13i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}