# Properties

 Label 2-12e2-144.85-c1-0-12 Degree $2$ Conductor $144$ Sign $0.953 - 0.300i$ 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 + (1.36 − 0.366i)2-s + 1.73i·3-s + (1.73 − i)4-s + (0.5 − 0.133i)5-s + (0.633 + 2.36i)6-s + (−2.13 + 1.23i)7-s + (1.99 − 2i)8-s − 2.99·9-s + (0.633 − 0.366i)10-s + (0.133 − 0.5i)11-s + (1.73 + 2.99i)12-s + (−1.23 − 4.59i)13-s + (−2.46 + 2.46i)14-s + (0.232 + 0.866i)15-s + (1.99 − 3.46i)16-s + 4·17-s + ⋯
 L(s)  = 1 + (0.965 − 0.258i)2-s + 0.999i·3-s + (0.866 − 0.5i)4-s + (0.223 − 0.0599i)5-s + (0.258 + 0.965i)6-s + (−0.806 + 0.465i)7-s + (0.707 − 0.707i)8-s − 0.999·9-s + (0.200 − 0.115i)10-s + (0.0403 − 0.150i)11-s + (0.499 + 0.866i)12-s + (−0.341 − 1.27i)13-s + (−0.658 + 0.658i)14-s + (0.0599 + 0.223i)15-s + (0.499 − 0.866i)16-s + 0.970·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.953 - 0.300i$ Analytic conductor: $$1.14984$$ Root analytic conductor: $$1.07230$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{144} (85, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ 0.953 - 0.300i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.75241 + 0.269722i$$ $$L(\frac12)$$ $$\approx$$ $$1.75241 + 0.269722i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.36 + 0.366i)T$$
3 $$1 - 1.73iT$$
good5 $$1 + (-0.5 + 0.133i)T + (4.33 - 2.5i)T^{2}$$
7 $$1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (-0.133 + 0.5i)T + (-9.52 - 5.5i)T^{2}$$
13 $$1 + (1.23 + 4.59i)T + (-11.2 + 6.5i)T^{2}$$
17 $$1 - 4T + 17T^{2}$$
19 $$1 + (3 - 3i)T - 19iT^{2}$$
23 $$1 + (-0.401 - 0.232i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (3.23 + 0.866i)T + (25.1 + 14.5i)T^{2}$$
31 $$1 + (-0.598 + 1.03i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (7.73 + 7.73i)T + 37iT^{2}$$
41 $$1 + (-9.69 - 5.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (2.33 - 8.69i)T + (-37.2 - 21.5i)T^{2}$$
47 $$1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-2.26 - 2.26i)T + 53iT^{2}$$
59 $$1 + (-5.59 + 1.5i)T + (51.0 - 29.5i)T^{2}$$
61 $$1 + (-14.4 - 3.86i)T + (52.8 + 30.5i)T^{2}$$
67 $$1 + (0.330 + 1.23i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + 10.9iT - 71T^{2}$$
73 $$1 + 0.535iT - 73T^{2}$$
79 $$1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (11.7 + 3.16i)T + (71.8 + 41.5i)T^{2}$$
89 $$1 - 11.8iT - 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}$$