# Properties

 Label 2-12e2-144.61-c1-0-5 Degree $2$ Conductor $144$ Sign $0.737 - 0.675i$ Analytic cond. $1.14984$ Root an. cond. $1.07230$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)2-s + 1.73·3-s − 2i·4-s + (−1.86 − 0.5i)5-s + (−1.73 + 1.73i)6-s + (3.86 + 2.23i)7-s + (2 + 2i)8-s + 2.99·9-s + (2.36 − 1.36i)10-s + (−0.5 − 1.86i)11-s − 3.46i·12-s + (−0.598 + 2.23i)13-s + (−6.09 + 1.63i)14-s + (−3.23 − 0.866i)15-s − 4·16-s + 4·17-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)2-s + 1.00·3-s − i·4-s + (−0.834 − 0.223i)5-s + (−0.707 + 0.707i)6-s + (1.46 + 0.843i)7-s + (0.707 + 0.707i)8-s + 0.999·9-s + (0.748 − 0.431i)10-s + (−0.150 − 0.562i)11-s − 0.999i·12-s + (−0.165 + 0.619i)13-s + (−1.62 + 0.436i)14-s + (−0.834 − 0.223i)15-s − 16-s + 0.970·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.737 - 0.675i$ Analytic conductor: $$1.14984$$ Root analytic conductor: $$1.07230$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{144} (61, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ 0.737 - 0.675i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.988813 + 0.384528i$$ $$L(\frac12)$$ $$\approx$$ $$0.988813 + 0.384528i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1 - i)T$$
3 $$1 - 1.73T$$
good5 $$1 + (1.86 + 0.5i)T + (4.33 + 2.5i)T^{2}$$
7 $$1 + (-3.86 - 2.23i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (0.5 + 1.86i)T + (-9.52 + 5.5i)T^{2}$$
13 $$1 + (0.598 - 2.23i)T + (-11.2 - 6.5i)T^{2}$$
17 $$1 - 4T + 17T^{2}$$
19 $$1 + (3 + 3i)T + 19iT^{2}$$
23 $$1 + (5.59 - 3.23i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (0.866 - 0.232i)T + (25.1 - 14.5i)T^{2}$$
31 $$1 + (4.59 + 7.96i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (4.26 - 4.26i)T - 37iT^{2}$$
41 $$1 + (-0.696 + 0.401i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (1.69 + 6.33i)T + (-37.2 + 21.5i)T^{2}$$
47 $$1 + (0.598 - 1.03i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-5.73 + 5.73i)T - 53iT^{2}$$
59 $$1 + (1.5 + 0.401i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (2.13 - 0.571i)T + (52.8 - 30.5i)T^{2}$$
67 $$1 + (2.23 - 8.33i)T + (-58.0 - 33.5i)T^{2}$$
71 $$1 + 2.92iT - 71T^{2}$$
73 $$1 - 7.46iT - 73T^{2}$$
79 $$1 + (0.866 - 1.5i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (14.1 - 3.79i)T + (71.8 - 41.5i)T^{2}$$
89 $$1 - 15.8iT - 89T^{2}$$
97 $$1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$