# Properties

 Label 2-43-43.42-c8-0-12 Degree $2$ Conductor $43$ Sign $-0.965 - 0.259i$ Analytic cond. $17.5172$ Root an. cond. $4.18536$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.87i·2-s + 114. i·3-s + 221.·4-s + 623. i·5-s − 672.·6-s + 2.43e3i·7-s + 2.80e3i·8-s − 6.52e3·9-s − 3.66e3·10-s + 2.48e4·11-s + 2.53e4i·12-s + 8.70e3·13-s − 1.43e4·14-s − 7.13e4·15-s + 4.02e4·16-s + 1.01e4·17-s + ⋯
 L(s)  = 1 + 0.367i·2-s + 1.41i·3-s + 0.865·4-s + 0.997i·5-s − 0.518·6-s + 1.01i·7-s + 0.684i·8-s − 0.994·9-s − 0.366·10-s + 1.69·11-s + 1.22i·12-s + 0.304·13-s − 0.372·14-s − 1.40·15-s + 0.613·16-s + 0.121·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-0.965 - 0.259i$ Analytic conductor: $$17.5172$$ Root analytic conductor: $$4.18536$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{43} (42, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :4),\ -0.965 - 0.259i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.328906 + 2.49172i$$ $$L(\frac12)$$ $$\approx$$ $$0.328906 + 2.49172i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (3.30e6 + 8.87e5i)T$$
good2 $$1 - 5.87iT - 256T^{2}$$
3 $$1 - 114. iT - 6.56e3T^{2}$$
5 $$1 - 623. iT - 3.90e5T^{2}$$
7 $$1 - 2.43e3iT - 5.76e6T^{2}$$
11 $$1 - 2.48e4T + 2.14e8T^{2}$$
13 $$1 - 8.70e3T + 8.15e8T^{2}$$
17 $$1 - 1.01e4T + 6.97e9T^{2}$$
19 $$1 + 1.86e5iT - 1.69e10T^{2}$$
23 $$1 + 3.87e5T + 7.83e10T^{2}$$
29 $$1 + 1.03e6iT - 5.00e11T^{2}$$
31 $$1 - 1.46e5T + 8.52e11T^{2}$$
37 $$1 + 2.77e6iT - 3.51e12T^{2}$$
41 $$1 - 8.90e4T + 7.98e12T^{2}$$
47 $$1 - 8.67e5T + 2.38e13T^{2}$$
53 $$1 - 1.05e7T + 6.22e13T^{2}$$
59 $$1 - 1.99e7T + 1.46e14T^{2}$$
61 $$1 + 9.32e6iT - 1.91e14T^{2}$$
67 $$1 + 1.83e7T + 4.06e14T^{2}$$
71 $$1 - 8.73e6iT - 6.45e14T^{2}$$
73 $$1 - 3.39e7iT - 8.06e14T^{2}$$
79 $$1 - 2.74e7T + 1.51e15T^{2}$$
83 $$1 + 6.16e7T + 2.25e15T^{2}$$
89 $$1 - 9.46e6iT - 3.93e15T^{2}$$
97 $$1 + 1.54e8T + 7.83e15T^{2}$$
show less
$$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

−15.06168142416800886475009611882, −14.23901559586935414316823590532, −11.83849658159973372705515853037, −11.17185330295453643004439427955, −9.934012552397659928931021558708, −8.758452190040863813088443249535, −6.85372973165241659269151208239, −5.75815312585002818748775113808, −3.89437542108192889005730631123, −2.44737678131276586809624688228, 1.09069196321194895350632066452, 1.54982406239232026318982965904, 3.82131811588116282389548212016, 6.20837768675777500843205357795, 7.15173129648073930957822884206, 8.390073378143182988985260794550, 10.15288939768821666071277237995, 11.79451656235787259295035898550, 12.27925267670217755582650935617, 13.41285181486887221899034808161