# Properties

 Degree $2$ Conductor $43$ Sign $0.656 + 0.754i$ Motivic weight $9$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.98 + 7.50i)2-s + (−64.3 + 9.70i)3-s + (93.4 + 409. i)4-s + (−1.64e3 + 1.12e3i)5-s + (312. − 541. i)6-s + (−2.49e3 − 4.32e3i)7-s + (−8.05e3 − 3.88e3i)8-s + (−1.47e4 + 4.55e3i)9-s + (1.42e3 − 1.90e4i)10-s + (−4.49e3 + 1.96e4i)11-s + (−9.98e3 − 2.54e4i)12-s + (5.96e3 + 7.95e4i)13-s + (4.74e4 + 7.14e3i)14-s + (9.49e4 − 8.81e4i)15-s + (−1.16e5 + 5.60e4i)16-s + (1.56e5 + 1.06e5i)17-s + ⋯
 L(s)  = 1 + (−0.264 + 0.331i)2-s + (−0.458 + 0.0691i)3-s + (0.182 + 0.799i)4-s + (−1.17 + 0.802i)5-s + (0.0983 − 0.170i)6-s + (−0.393 − 0.681i)7-s + (−0.695 − 0.334i)8-s + (−0.749 + 0.231i)9-s + (0.0451 − 0.602i)10-s + (−0.0924 + 0.405i)11-s + (−0.138 − 0.354i)12-s + (0.0578 + 0.772i)13-s + (0.330 + 0.0497i)14-s + (0.484 − 0.449i)15-s + (−0.443 + 0.213i)16-s + (0.454 + 0.309i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.656 + 0.754i$ Motivic weight: $$9$$ Character: $\chi_{43} (24, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :9/2),\ 0.656 + 0.754i)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$0.109616 - 0.0499431i$$ $$L(\frac12)$$ $$\approx$$ $$0.109616 - 0.0499431i$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (1.78e7 - 1.35e7i)T$$
good2 $$1 + (5.98 - 7.50i)T + (-113. - 499. i)T^{2}$$
3 $$1 + (64.3 - 9.70i)T + (1.88e4 - 5.80e3i)T^{2}$$
5 $$1 + (1.64e3 - 1.12e3i)T + (7.13e5 - 1.81e6i)T^{2}$$
7 $$1 + (2.49e3 + 4.32e3i)T + (-2.01e7 + 3.49e7i)T^{2}$$
11 $$1 + (4.49e3 - 1.96e4i)T + (-2.12e9 - 1.02e9i)T^{2}$$
13 $$1 + (-5.96e3 - 7.95e4i)T + (-1.04e10 + 1.58e9i)T^{2}$$
17 $$1 + (-1.56e5 - 1.06e5i)T + (4.33e10 + 1.10e11i)T^{2}$$
19 $$1 + (2.48e4 + 7.66e3i)T + (2.66e11 + 1.81e11i)T^{2}$$
23 $$1 + (4.94e5 + 4.58e5i)T + (1.34e11 + 1.79e12i)T^{2}$$
29 $$1 + (-6.85e6 - 1.03e6i)T + (1.38e13 + 4.27e12i)T^{2}$$
31 $$1 + (1.30e6 + 3.31e6i)T + (-1.93e13 + 1.79e13i)T^{2}$$
37 $$1 + (7.80e6 - 1.35e7i)T + (-6.49e13 - 1.12e14i)T^{2}$$
41 $$1 + (-1.93e7 + 2.42e7i)T + (-7.28e13 - 3.19e14i)T^{2}$$
47 $$1 + (3.43e6 + 1.50e7i)T + (-1.00e15 + 4.85e14i)T^{2}$$
53 $$1 + (-1.06e6 + 1.42e7i)T + (-3.26e15 - 4.91e14i)T^{2}$$
59 $$1 + (1.25e8 - 6.02e7i)T + (5.40e15 - 6.77e15i)T^{2}$$
61 $$1 + (6.11e7 - 1.55e8i)T + (-8.57e15 - 7.95e15i)T^{2}$$
67 $$1 + (3.10e7 + 9.57e6i)T + (2.24e16 + 1.53e16i)T^{2}$$
71 $$1 + (-6.74e7 + 6.26e7i)T + (3.42e15 - 4.57e16i)T^{2}$$
73 $$1 + (-1.43e6 - 1.90e7i)T + (-5.82e16 + 8.77e15i)T^{2}$$
79 $$1 + (-6.87e7 - 1.19e8i)T + (-5.99e16 + 1.03e17i)T^{2}$$
83 $$1 + (7.94e8 - 1.19e8i)T + (1.78e17 - 5.51e16i)T^{2}$$
89 $$1 + (-7.74e8 + 1.16e8i)T + (3.34e17 - 1.03e17i)T^{2}$$
97 $$1 + (-2.08e8 + 9.11e8i)T + (-6.84e17 - 3.29e17i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$