# Properties

 Label 2-43-43.36-c5-0-3 Degree $2$ Conductor $43$ Sign $0.803 - 0.595i$ Analytic cond. $6.89650$ Root an. cond. $2.62611$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 10.9·2-s + (8.65 + 14.9i)3-s + 88.5·4-s + (−5.02 − 8.70i)5-s + (−94.9 − 164. i)6-s + (36.4 − 63.0i)7-s − 620.·8-s + (−28.1 + 48.8i)9-s + (55.1 + 95.5i)10-s + 500.·11-s + (766. + 1.32e3i)12-s + (376. − 652. i)13-s + (−399. + 692. i)14-s + (86.9 − 150. i)15-s + 3.98e3·16-s + (−829. + 1.43e3i)17-s + ⋯
 L(s)  = 1 − 1.94·2-s + (0.554 + 0.961i)3-s + 2.76·4-s + (−0.0898 − 0.155i)5-s + (−1.07 − 1.86i)6-s + (0.280 − 0.486i)7-s − 3.43·8-s + (−0.115 + 0.200i)9-s + (0.174 + 0.302i)10-s + 1.24·11-s + (1.53 + 2.66i)12-s + (0.618 − 1.07i)13-s + (−0.545 + 0.944i)14-s + (0.0997 − 0.172i)15-s + 3.89·16-s + (−0.695 + 1.20i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.803 - 0.595i$ Analytic conductor: $$6.89650$$ Root analytic conductor: $$2.62611$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{43} (36, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :5/2),\ 0.803 - 0.595i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.867489 + 0.286633i$$ $$L(\frac12)$$ $$\approx$$ $$0.867489 + 0.286633i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (1.20e4 + 1.25e3i)T$$
good2 $$1 + 10.9T + 32T^{2}$$
3 $$1 + (-8.65 - 14.9i)T + (-121.5 + 210. i)T^{2}$$
5 $$1 + (5.02 + 8.70i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 + (-36.4 + 63.0i)T + (-8.40e3 - 1.45e4i)T^{2}$$
11 $$1 - 500.T + 1.61e5T^{2}$$
13 $$1 + (-376. + 652. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + (829. - 1.43e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-1.13e3 - 1.95e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-20.0 - 34.8i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (638. - 1.10e3i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 + (-2.82e3 - 4.89e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (-1.52e3 - 2.64e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 2.61e3T + 1.15e8T^{2}$$
47 $$1 - 2.10e4T + 2.29e8T^{2}$$
53 $$1 + (1.70e4 + 2.95e4i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 - 1.74e4T + 7.14e8T^{2}$$
61 $$1 + (-448. + 776. i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.05e4 + 3.56e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + (-1.09e4 + 1.89e4i)T + (-9.02e8 - 1.56e9i)T^{2}$$
73 $$1 + (1.94e4 - 3.36e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-4.59e4 + 7.96e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + (-3.91e4 - 6.78e4i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + (2.84e4 + 4.92e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + 331.T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$