# Properties

 Label 2-43-43.19-c6-0-1 Degree $2$ Conductor $43$ Sign $0.954 - 0.298i$ 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 + (−6.36 − 5.07i)2-s + (−3.86 − 25.6i)3-s + (0.504 + 2.21i)4-s + (85.2 + 125. i)5-s + (−105. + 182. i)6-s + (−386. + 223. i)7-s + (−218. + 452. i)8-s + (53.1 − 16.3i)9-s + (92.0 − 1.22e3i)10-s + (244. − 1.06e3i)11-s + (54.8 − 21.5i)12-s + (308. + 4.12e3i)13-s + (3.59e3 + 541. i)14-s + (2.87e3 − 2.67e3i)15-s + (3.81e3 − 1.83e3i)16-s + (631. + 430. i)17-s + ⋯
 L(s)  = 1 + (−0.795 − 0.634i)2-s + (−0.143 − 0.950i)3-s + (0.00789 + 0.0345i)4-s + (0.681 + 1.00i)5-s + (−0.489 + 0.846i)6-s + (−1.12 + 0.650i)7-s + (−0.425 + 0.884i)8-s + (0.0728 − 0.0224i)9-s + (0.0920 − 1.22i)10-s + (0.183 − 0.803i)11-s + (0.0317 − 0.0124i)12-s + (0.140 + 1.87i)13-s + (1.30 + 0.197i)14-s + (0.852 − 0.791i)15-s + (0.931 − 0.448i)16-s + (0.128 + 0.0875i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.954 - 0.298i$ Analytic conductor: $$9.89232$$ Root analytic conductor: $$3.14520$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{43} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :3),\ 0.954 - 0.298i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.787390 + 0.120441i$$ $$L(\frac12)$$ $$\approx$$ $$0.787390 + 0.120441i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-4.89e4 - 6.26e4i)T$$
good2 $$1 + (6.36 + 5.07i)T + (14.2 + 62.3i)T^{2}$$
3 $$1 + (3.86 + 25.6i)T + (-696. + 214. i)T^{2}$$
5 $$1 + (-85.2 - 125. i)T + (-5.70e3 + 1.45e4i)T^{2}$$
7 $$1 + (386. - 223. i)T + (5.88e4 - 1.01e5i)T^{2}$$
11 $$1 + (-244. + 1.06e3i)T + (-1.59e6 - 7.68e5i)T^{2}$$
13 $$1 + (-308. - 4.12e3i)T + (-4.77e6 + 7.19e5i)T^{2}$$
17 $$1 + (-631. - 430. i)T + (8.81e6 + 2.24e7i)T^{2}$$
19 $$1 + (-556. + 1.80e3i)T + (-3.88e7 - 2.65e7i)T^{2}$$
23 $$1 + (-6.11e3 - 5.67e3i)T + (1.10e7 + 1.47e8i)T^{2}$$
29 $$1 + (1.60e3 - 1.06e4i)T + (-5.68e8 - 1.75e8i)T^{2}$$
31 $$1 + (-1.07e4 - 2.75e4i)T + (-6.50e8 + 6.03e8i)T^{2}$$
37 $$1 + (4.49e4 + 2.59e4i)T + (1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + (6.73e4 - 8.44e4i)T + (-1.05e9 - 4.63e9i)T^{2}$$
47 $$1 + (-2.79e4 - 1.22e5i)T + (-9.71e9 + 4.67e9i)T^{2}$$
53 $$1 + (1.20e4 - 1.61e5i)T + (-2.19e10 - 3.30e9i)T^{2}$$
59 $$1 + (1.83e4 - 8.85e3i)T + (2.62e10 - 3.29e10i)T^{2}$$
61 $$1 + (-3.73e5 - 1.46e5i)T + (3.77e10 + 3.50e10i)T^{2}$$
67 $$1 + (-3.38e5 - 1.04e5i)T + (7.47e10 + 5.09e10i)T^{2}$$
71 $$1 + (4.21e5 + 4.54e5i)T + (-9.57e9 + 1.27e11i)T^{2}$$
73 $$1 + (-4.09e4 + 3.07e3i)T + (1.49e11 - 2.25e10i)T^{2}$$
79 $$1 + (2.96e5 + 5.13e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + (6.19e5 - 9.33e4i)T + (3.12e11 - 9.63e10i)T^{2}$$
89 $$1 + (-8.79e4 - 5.83e5i)T + (-4.74e11 + 1.46e11i)T^{2}$$
97 $$1 + (2.00e4 - 8.79e4i)T + (-7.50e11 - 3.61e11i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$