# Properties

 Label 2-42-7.5-c2-0-0 Degree $2$ Conductor $42$ Sign $0.784 - 0.620i$ Analytic cond. $1.14441$ Root an. cond. $1.06977$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (7.24 + 4.18i)5-s + 2.44i·6-s + (−6.74 + 1.88i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (−10.2 + 5.91i)10-s + (−3 − 5.19i)11-s + (−2.99 − 1.73i)12-s − 17.8i·13-s + (2.46 − 9.58i)14-s + 14.4·15-s + (−2.00 + 3.46i)16-s + (−16.2 + 9.37i)17-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (1.44 + 0.836i)5-s + 0.408i·6-s + (−0.963 + 0.268i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−1.02 + 0.591i)10-s + (−0.272 − 0.472i)11-s + (−0.249 − 0.144i)12-s − 1.37i·13-s + (0.176 − 0.684i)14-s + 0.965·15-s + (−0.125 + 0.216i)16-s + (−0.955 + 0.551i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$42$$    =    $$2 \cdot 3 \cdot 7$$ Sign: $0.784 - 0.620i$ Analytic conductor: $$1.14441$$ Root analytic conductor: $$1.06977$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{42} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 42,\ (\ :1),\ 0.784 - 0.620i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.05547 + 0.366754i$$ $$L(\frac12)$$ $$\approx$$ $$1.05547 + 0.366754i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 - 1.22i)T$$
3 $$1 + (-1.5 + 0.866i)T$$
7 $$1 + (6.74 - 1.88i)T$$
good5 $$1 + (-7.24 - 4.18i)T + (12.5 + 21.6i)T^{2}$$
11 $$1 + (3 + 5.19i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + 17.8iT - 169T^{2}$$
17 $$1 + (16.2 - 9.37i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (14.7 + 8.51i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (6.72 - 11.6i)T + (-264.5 - 458. i)T^{2}$$
29 $$1 - 33.9T + 841T^{2}$$
31 $$1 + (-12.7 + 7.37i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + (2.98 - 5.17i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 - 35.2iT - 1.68e3T^{2}$$
43 $$1 - 15.4T + 1.84e3T^{2}$$
47 $$1 + (28.7 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (-17.2 - 29.9i)T + (-1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (-23.6 + 13.6i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (34.9 + 20.1i)T + (1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (-57.1 - 99.0i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 18.6T + 5.04e3T^{2}$$
73 $$1 + (101. - 58.5i)T + (2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (-44.1 + 76.5i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 - 75.7iT - 6.88e3T^{2}$$
89 $$1 + (18 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 - 30.5iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$