# Properties

 Label 2-43-43.37-c6-0-8 Degree $2$ Conductor $43$ Sign $0.915 - 0.402i$ Analytic cond. $9.89232$ Root an. cond. $3.14520$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.64i·2-s + (−0.554 − 0.320i)3-s + 57.0·4-s + (14.6 + 8.48i)5-s + (0.846 − 1.46i)6-s + (131. − 76.0i)7-s + 320. i·8-s + (−364. − 630. i)9-s + (−22.4 + 38.8i)10-s + 1.69e3·11-s + (−31.6 − 18.2i)12-s + (892. + 1.54e3i)13-s + (201. + 348. i)14-s + (−5.43 − 9.40i)15-s + 2.80e3·16-s + (1.21e3 + 2.09e3i)17-s + ⋯
 L(s)  = 1 + 0.330i·2-s + (−0.0205 − 0.0118i)3-s + 0.890·4-s + (0.117 + 0.0678i)5-s + (0.00392 − 0.00678i)6-s + (0.384 − 0.221i)7-s + 0.625i·8-s + (−0.499 − 0.865i)9-s + (−0.0224 + 0.0388i)10-s + 1.27·11-s + (−0.0182 − 0.0105i)12-s + (0.406 + 0.703i)13-s + (0.0733 + 0.127i)14-s + (−0.00160 − 0.00278i)15-s + 0.684·16-s + (0.246 + 0.426i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.915 - 0.402i$ Analytic conductor: $$9.89232$$ Root analytic conductor: $$3.14520$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{43} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :3),\ 0.915 - 0.402i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$2.25585 + 0.474136i$$ $$L(\frac12)$$ $$\approx$$ $$2.25585 + 0.474136i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (3.81e4 + 6.97e4i)T$$
good2 $$1 - 2.64iT - 64T^{2}$$
3 $$1 + (0.554 + 0.320i)T + (364.5 + 631. i)T^{2}$$
5 $$1 + (-14.6 - 8.48i)T + (7.81e3 + 1.35e4i)T^{2}$$
7 $$1 + (-131. + 76.0i)T + (5.88e4 - 1.01e5i)T^{2}$$
11 $$1 - 1.69e3T + 1.77e6T^{2}$$
13 $$1 + (-892. - 1.54e3i)T + (-2.41e6 + 4.18e6i)T^{2}$$
17 $$1 + (-1.21e3 - 2.09e3i)T + (-1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-8.14e3 - 4.70e3i)T + (2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (2.99e3 - 5.19e3i)T + (-7.40e7 - 1.28e8i)T^{2}$$
29 $$1 + (-3.63e3 + 2.09e3i)T + (2.97e8 - 5.15e8i)T^{2}$$
31 $$1 + (-1.17e4 + 2.03e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (4.41e4 + 2.54e4i)T + (1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 6.83e4T + 4.75e9T^{2}$$
47 $$1 + 1.43e5T + 1.07e10T^{2}$$
53 $$1 + (-4.02e3 + 6.97e3i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 - 2.70e5T + 4.21e10T^{2}$$
61 $$1 + (-1.95e5 + 1.12e5i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (6.97e4 - 1.20e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + (5.01e5 - 2.89e5i)T + (6.40e10 - 1.10e11i)T^{2}$$
73 $$1 + (-3.57e5 + 2.06e5i)T + (7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (3.87e5 + 6.71e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + (2.34e5 - 4.06e5i)T + (-1.63e11 - 2.83e11i)T^{2}$$
89 $$1 + (5.11e5 + 2.95e5i)T + (2.48e11 + 4.30e11i)T^{2}$$
97 $$1 + 1.06e6T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$