# Properties

 Degree $2$ Conductor $42$ Sign $0.452 + 0.891i$ 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 − 5.91i·5-s + 2.44i·6-s + (6.24 − 3.16i)7-s − 2.82·8-s − 2.99·9-s + 8.36i·10-s − 1.75·11-s − 3.46i·12-s + 18.7i·13-s + (−8.82 + 4.47i)14-s − 10.2·15-s + 4.00·16-s + 23.4i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577i·3-s + 0.500·4-s − 1.18i·5-s + 0.408i·6-s + (0.891 − 0.452i)7-s − 0.353·8-s − 0.333·9-s + 0.836i·10-s − 0.159·11-s − 0.288i·12-s + 1.44i·13-s + (−0.630 + 0.319i)14-s − 0.682·15-s + 0.250·16-s + 1.38i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$42$$    =    $$2 \cdot 3 \cdot 7$$ Sign: $0.452 + 0.891i$ Motivic weight: $$2$$ Character: $\chi_{42} (13, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 42,\ (\ :1),\ 0.452 + 0.891i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.715425 - 0.439281i$$ $$L(\frac12)$$ $$\approx$$ $$0.715425 - 0.439281i$$ $$L(2)$$ 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 + (-6.24 + 3.16i)T$$
good5 $$1 + 5.91iT - 25T^{2}$$
11 $$1 + 1.75T + 121T^{2}$$
13 $$1 - 18.7iT - 169T^{2}$$
17 $$1 - 23.4iT - 289T^{2}$$
19 $$1 + 23.0iT - 361T^{2}$$
23 $$1 - 18.7T + 529T^{2}$$
29 $$1 - 30T + 841T^{2}$$
31 $$1 - 8.60iT - 961T^{2}$$
37 $$1 + 70.9T + 1.36e3T^{2}$$
41 $$1 - 41.3iT - 1.68e3T^{2}$$
43 $$1 - 10.4T + 1.84e3T^{2}$$
47 $$1 - 38.6iT - 2.20e3T^{2}$$
53 $$1 + 37.0T + 2.80e3T^{2}$$
59 $$1 + 97.4iT - 3.48e3T^{2}$$
61 $$1 - 16.7iT - 3.72e3T^{2}$$
67 $$1 - 60.9T + 4.48e3T^{2}$$
71 $$1 + 110.T + 5.04e3T^{2}$$
73 $$1 + 56.7iT - 5.32e3T^{2}$$
79 $$1 + 69.8T + 6.24e3T^{2}$$
83 $$1 - 6.43iT - 6.88e3T^{2}$$
89 $$1 + 42.0iT - 7.92e3T^{2}$$
97 $$1 - 51.7iT - 9.40e3T^{2}$$
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$$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

−15.90934900974497686646059791775, −14.39871840141845999389564390836, −13.13656549053271463642757229727, −11.97128720903096030219212174477, −10.83891403603784171011298473820, −9.060027996856958197971697759825, −8.240003287991511385307968377459, −6.78491589255249120465243156852, −4.76563063945721265405188761708, −1.47387573767770102315147410442, 2.93119924689143621118693881153, 5.45319462803678424647293376295, 7.27641463192859787441316648427, 8.545921331359559039305387495455, 10.13599273426427390814173032677, 10.86519446483954773602597802227, 12.04786611677819163528723866998, 14.10268182626640767903968524680, 15.05042644296659737085031902424, 15.85132384144099200782981461510