# Properties

 Label 2-72-24.5-c2-0-4 Degree $2$ Conductor $72$ Sign $0.731 - 0.681i$ Analytic cond. $1.96185$ Root an. cond. $1.40066$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.77 + 0.921i)2-s + (2.30 + 3.27i)4-s − 1.07·5-s + 7.21·7-s + (1.07 + 7.92i)8-s + (−1.90 − 0.990i)10-s − 16.3·11-s − 21.6i·13-s + (12.8 + 6.64i)14-s + (−5.39 + 15.0i)16-s − 18.9i·17-s + 17.0i·19-s + (−2.47 − 3.51i)20-s + (−29.0 − 15.0i)22-s − 1.11i·23-s + ⋯
 L(s)  = 1 + (0.887 + 0.460i)2-s + (0.575 + 0.817i)4-s − 0.214·5-s + 1.03·7-s + (0.134 + 0.990i)8-s + (−0.190 − 0.0990i)10-s − 1.48·11-s − 1.66i·13-s + (0.914 + 0.474i)14-s + (−0.337 + 0.941i)16-s − 1.11i·17-s + 0.897i·19-s + (−0.123 − 0.175i)20-s + (−1.31 − 0.684i)22-s − 0.0485i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$72$$    =    $$2^{3} \cdot 3^{2}$$ Sign: $0.731 - 0.681i$ Analytic conductor: $$1.96185$$ Root analytic conductor: $$1.40066$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{72} (53, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 72,\ (\ :1),\ 0.731 - 0.681i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.80103 + 0.709206i$$ $$L(\frac12)$$ $$\approx$$ $$1.80103 + 0.709206i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.77 - 0.921i)T$$
3 $$1$$
good5 $$1 + 1.07T + 25T^{2}$$
7 $$1 - 7.21T + 49T^{2}$$
11 $$1 + 16.3T + 121T^{2}$$
13 $$1 + 21.6iT - 169T^{2}$$
17 $$1 + 18.9iT - 289T^{2}$$
19 $$1 - 17.0iT - 361T^{2}$$
23 $$1 + 1.11iT - 529T^{2}$$
29 $$1 - 29.4T + 841T^{2}$$
31 $$1 - 5.63T + 961T^{2}$$
37 $$1 + 17.0iT - 1.36e3T^{2}$$
41 $$1 - 27.4iT - 1.68e3T^{2}$$
43 $$1 - 52.3iT - 1.84e3T^{2}$$
47 $$1 - 64.5iT - 2.20e3T^{2}$$
53 $$1 - 35.9T + 2.80e3T^{2}$$
59 $$1 - 56.8T + 3.48e3T^{2}$$
61 $$1 - 69.3iT - 3.72e3T^{2}$$
67 $$1 + 69.3iT - 4.48e3T^{2}$$
71 $$1 + 98.4iT - 5.04e3T^{2}$$
73 $$1 - 37.6T + 5.32e3T^{2}$$
79 $$1 + 127.T + 6.24e3T^{2}$$
83 $$1 - 7.75T + 6.88e3T^{2}$$
89 $$1 + 76.1iT - 7.92e3T^{2}$$
97 $$1 - 4.84T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$