# Properties

 Label 2-12e2-36.23-c1-0-5 Degree $2$ Conductor $144$ Sign $-0.984 + 0.173i$ Analytic cond. $1.14984$ Root an. cond. $1.07230$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 + 0.866i)3-s + (−3 − 1.73i)5-s + (−3 + 1.73i)7-s + (1.5 − 2.59i)9-s + (−1.5 − 2.59i)11-s + (−2 + 3.46i)13-s + 6·15-s + 1.73i·17-s − 1.73i·19-s + (3 − 5.19i)21-s + (3.5 + 6.06i)25-s + 5.19i·27-s + (−3 + 1.73i)29-s + (4.5 + 2.59i)33-s + 12·35-s + ⋯
 L(s)  = 1 + (−0.866 + 0.499i)3-s + (−1.34 − 0.774i)5-s + (−1.13 + 0.654i)7-s + (0.5 − 0.866i)9-s + (−0.452 − 0.783i)11-s + (−0.554 + 0.960i)13-s + 1.54·15-s + 0.420i·17-s − 0.397i·19-s + (0.654 − 1.13i)21-s + (0.700 + 1.21i)25-s + 0.999i·27-s + (−0.557 + 0.321i)29-s + (0.783 + 0.452i)33-s + 2.02·35-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $-0.984 + 0.173i$ Analytic conductor: $$1.14984$$ Root analytic conductor: $$1.07230$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{144} (95, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ -0.984 + 0.173i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (1.5 - 0.866i)T$$
good5 $$1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2}$$
7 $$1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 - 1.73iT - 17T^{2}$$
19 $$1 + 1.73iT - 19T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3 - 1.73i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (15.5 + 26.8i)T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-4.5 + 2.59i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + 11T + 73T^{2}$$
79 $$1 + (-3 + 1.73i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 13.8iT - 89T^{2}$$
97 $$1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$