# Properties

 Degree $2$ Conductor $42$ Sign $0.947 + 0.320i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 1.01i·5-s − 2.44i·6-s + (−2.24 + 6.63i)7-s + 2.82·8-s − 2.99·9-s − 1.43i·10-s − 10.2·11-s − 3.46i·12-s + 8.95i·13-s + (−3.17 + 9.37i)14-s − 1.75·15-s + 4.00·16-s − 30.4i·17-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.202i·5-s − 0.408i·6-s + (−0.320 + 0.947i)7-s + 0.353·8-s − 0.333·9-s − 0.143i·10-s − 0.931·11-s − 0.288i·12-s + 0.689i·13-s + (−0.226 + 0.669i)14-s − 0.117·15-s + 0.250·16-s − 1.78i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$42$$    =    $$2 \cdot 3 \cdot 7$$ Sign: $0.947 + 0.320i$ Motivic weight: $$2$$ Character: $\chi_{42} (13, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 42,\ (\ :1),\ 0.947 + 0.320i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.44507 - 0.237750i$$ $$L(\frac12)$$ $$\approx$$ $$1.44507 - 0.237750i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.41T$$
3 $$1 + 1.73iT$$
7 $$1 + (2.24 - 6.63i)T$$
good5 $$1 + 1.01iT - 25T^{2}$$
11 $$1 + 10.2T + 121T^{2}$$
13 $$1 - 8.95iT - 169T^{2}$$
17 $$1 + 30.4iT - 289T^{2}$$
19 $$1 - 16.1iT - 361T^{2}$$
23 $$1 + 6.72T + 529T^{2}$$
29 $$1 - 30T + 841T^{2}$$
31 $$1 + 50.1iT - 961T^{2}$$
37 $$1 - 30.9T + 1.36e3T^{2}$$
41 $$1 - 7.10iT - 1.68e3T^{2}$$
43 $$1 + 74.4T + 1.84e3T^{2}$$
47 $$1 - 58.2iT - 2.20e3T^{2}$$
53 $$1 + 70.9T + 2.80e3T^{2}$$
59 $$1 - 0.492iT - 3.48e3T^{2}$$
61 $$1 + 2.86iT - 3.72e3T^{2}$$
67 $$1 - 27.0T + 4.48e3T^{2}$$
71 $$1 - 50.6T + 5.04e3T^{2}$$
73 $$1 - 70.6iT - 5.32e3T^{2}$$
79 $$1 - 133.T + 6.24e3T^{2}$$
83 $$1 - 104. iT - 6.88e3T^{2}$$
89 $$1 + 144. iT - 7.92e3T^{2}$$
97 $$1 - 100. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$