# Properties

 Degree 2 Conductor 43 Sign $0.292 + 0.956i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.32 + 8.97i)2-s + (20.4 − 42.4i)3-s + (−22.0 + 27.6i)4-s + (−183. − 41.9i)5-s + 469.·6-s − 28.9i·7-s + (278. + 63.5i)8-s + (−927. − 1.16e3i)9-s + (−417. − 1.83e3i)10-s + (−675. − 846. i)11-s + (721. + 1.49e3i)12-s + (485. − 2.12e3i)13-s + (260. − 125. i)14-s + (−5.53e3 + 6.93e3i)15-s + (1.13e3 + 4.98e3i)16-s + (−1.87e3 − 8.19e3i)17-s + ⋯
 L(s)  = 1 + (0.540 + 1.12i)2-s + (0.756 − 1.57i)3-s + (−0.344 + 0.431i)4-s + (−1.46 − 0.335i)5-s + 2.17·6-s − 0.0845i·7-s + (0.544 + 0.124i)8-s + (−1.27 − 1.59i)9-s + (−0.417 − 1.83i)10-s + (−0.507 − 0.636i)11-s + (0.417 + 0.867i)12-s + (0.221 − 0.968i)13-s + (0.0948 − 0.0456i)14-s + (−1.63 + 2.05i)15-s + (0.277 + 1.21i)16-s + (−0.380 − 1.66i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.292 + 0.956i$ motivic weight = $$6$$ character : $\chi_{43} (2, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ 0.292 + 0.956i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$1.68954 - 1.25041i$$ $$L(\frac12)$$ $$\approx$$ $$1.68954 - 1.25041i$$ $$L(4)$$ 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 + (-5.83e4 - 5.39e4i)T$$
good2 $$1 + (-4.32 - 8.97i)T + (-39.9 + 50.0i)T^{2}$$
3 $$1 + (-20.4 + 42.4i)T + (-454. - 569. i)T^{2}$$
5 $$1 + (183. + 41.9i)T + (1.40e4 + 6.77e3i)T^{2}$$
7 $$1 + 28.9iT - 1.17e5T^{2}$$
11 $$1 + (675. + 846. i)T + (-3.94e5 + 1.72e6i)T^{2}$$
13 $$1 + (-485. + 2.12e3i)T + (-4.34e6 - 2.09e6i)T^{2}$$
17 $$1 + (1.87e3 + 8.19e3i)T + (-2.17e7 + 1.04e7i)T^{2}$$
19 $$1 + (-6.87e3 - 5.48e3i)T + (1.04e7 + 4.58e7i)T^{2}$$
23 $$1 + (-7.55e3 - 9.46e3i)T + (-3.29e7 + 1.44e8i)T^{2}$$
29 $$1 + (-1.04e4 - 2.17e4i)T + (-3.70e8 + 4.65e8i)T^{2}$$
31 $$1 + (1.95e4 - 9.42e3i)T + (5.53e8 - 6.93e8i)T^{2}$$
37 $$1 + 1.73e4iT - 2.56e9T^{2}$$
41 $$1 + (-9.81e3 + 4.72e3i)T + (2.96e9 - 3.71e9i)T^{2}$$
47 $$1 + (-8.41e4 + 1.05e5i)T + (-2.39e9 - 1.05e10i)T^{2}$$
53 $$1 + (4.05e4 + 1.77e5i)T + (-1.99e10 + 9.61e9i)T^{2}$$
59 $$1 + (7.06e4 + 3.09e5i)T + (-3.80e10 + 1.83e10i)T^{2}$$
61 $$1 + (2.98e4 - 6.19e4i)T + (-3.21e10 - 4.02e10i)T^{2}$$
67 $$1 + (2.35e5 - 2.94e5i)T + (-2.01e10 - 8.81e10i)T^{2}$$
71 $$1 + (-3.85e5 - 3.07e5i)T + (2.85e10 + 1.24e11i)T^{2}$$
73 $$1 + (2.44e5 + 5.58e4i)T + (1.36e11 + 6.56e10i)T^{2}$$
79 $$1 - 4.19e5T + 2.43e11T^{2}$$
83 $$1 + (5.53e5 + 2.66e5i)T + (2.03e11 + 2.55e11i)T^{2}$$
89 $$1 + (-5.05e4 + 1.05e5i)T + (-3.09e11 - 3.88e11i)T^{2}$$
97 $$1 + (2.92e5 + 3.67e5i)T + (-1.85e11 + 8.12e11i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}