# Properties

 Label 2-43-43.7-c6-0-12 Degree $2$ Conductor $43$ Sign $0.349 + 0.936i$ Analytic cond. $9.89232$ Root an. cond. $3.14520$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + 6.20i·2-s + (−29.5 + 17.0i)3-s + 25.5·4-s + (−46.5 + 26.8i)5-s + (−105. − 183. i)6-s + (−448. − 259. i)7-s + 555. i·8-s + (218. − 378. i)9-s + (−166. − 288. i)10-s + 668.·11-s + (−754. + 435. i)12-s + (1.45e3 − 2.52e3i)13-s + (1.60e3 − 2.78e3i)14-s + (917. − 1.58e3i)15-s − 1.81e3·16-s + (4.62e3 − 8.00e3i)17-s + ⋯
 L(s)  = 1 + 0.775i·2-s + (−1.09 + 0.632i)3-s + 0.398·4-s + (−0.372 + 0.215i)5-s + (−0.490 − 0.849i)6-s + (−1.30 − 0.755i)7-s + 1.08i·8-s + (0.299 − 0.519i)9-s + (−0.166 − 0.288i)10-s + 0.502·11-s + (−0.436 + 0.252i)12-s + (0.664 − 1.15i)13-s + (0.585 − 1.01i)14-s + (0.271 − 0.471i)15-s − 0.442·16-s + (0.940 − 1.62i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.349 + 0.936i$ Analytic conductor: $$9.89232$$ Root analytic conductor: $$3.14520$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{43} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :3),\ 0.349 + 0.936i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.216780 - 0.150458i$$ $$L(\frac12)$$ $$\approx$$ $$0.216780 - 0.150458i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (7.66e4 - 2.12e4i)T$$
good2 $$1 - 6.20iT - 64T^{2}$$
3 $$1 + (29.5 - 17.0i)T + (364.5 - 631. i)T^{2}$$
5 $$1 + (46.5 - 26.8i)T + (7.81e3 - 1.35e4i)T^{2}$$
7 $$1 + (448. + 259. i)T + (5.88e4 + 1.01e5i)T^{2}$$
11 $$1 - 668.T + 1.77e6T^{2}$$
13 $$1 + (-1.45e3 + 2.52e3i)T + (-2.41e6 - 4.18e6i)T^{2}$$
17 $$1 + (-4.62e3 + 8.00e3i)T + (-1.20e7 - 2.09e7i)T^{2}$$
19 $$1 + (8.61e3 - 4.97e3i)T + (2.35e7 - 4.07e7i)T^{2}$$
23 $$1 + (-1.26e3 - 2.18e3i)T + (-7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + (2.89e4 + 1.67e4i)T + (2.97e8 + 5.15e8i)T^{2}$$
31 $$1 + (1.21e4 + 2.11e4i)T + (-4.43e8 + 7.68e8i)T^{2}$$
37 $$1 + (-2.02e4 + 1.16e4i)T + (1.28e9 - 2.22e9i)T^{2}$$
41 $$1 + 1.15e5T + 4.75e9T^{2}$$
47 $$1 - 6.17e4T + 1.07e10T^{2}$$
53 $$1 + (7.85e4 + 1.36e5i)T + (-1.10e10 + 1.91e10i)T^{2}$$
59 $$1 + 1.24e5T + 4.21e10T^{2}$$
61 $$1 + (-1.42e5 - 8.22e4i)T + (2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (1.53e5 + 2.66e5i)T + (-4.52e10 + 7.83e10i)T^{2}$$
71 $$1 + (-1.44e5 - 8.36e4i)T + (6.40e10 + 1.10e11i)T^{2}$$
73 $$1 + (9.21e4 + 5.32e4i)T + (7.56e10 + 1.31e11i)T^{2}$$
79 $$1 + (4.26e4 - 7.38e4i)T + (-1.21e11 - 2.10e11i)T^{2}$$
83 $$1 + (-2.15e5 - 3.73e5i)T + (-1.63e11 + 2.83e11i)T^{2}$$
89 $$1 + (-3.36e5 + 1.94e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 - 1.25e6T + 8.32e11T^{2}$$
show more
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−14.95869449976524323442637853324, −13.35974299121166828554029756482, −11.82839852610548466449393669243, −10.86040163420397750025396287986, −9.811488548174890189829300376740, −7.71775918588168451124942321848, −6.49322617002465495418837983780, −5.50302131151754846509552642770, −3.54987458622633281369572807999, −0.13848413660799481942449066795, 1.62527753598556205534325938878, 3.65468102228167398611673199722, 6.12415058937497287020714225275, 6.71649440468010969477700641205, 8.943036733313671440407000599440, 10.47403147427232446925352799660, 11.59999412229275568892861776564, 12.36515578804498137261559582158, 12.98058230055416195283117022160, 15.11598047362730338322164601825