# Properties

 Label 2-12e2-144.85-c1-0-13 Degree $2$ Conductor $144$ Sign $0.565 + 0.825i$ 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 + (−1.38 + 0.276i)2-s + (0.841 − 1.51i)3-s + (1.84 − 0.765i)4-s + (−0.0691 + 0.0185i)5-s + (−0.748 + 2.33i)6-s + (1.28 − 0.740i)7-s + (−2.35 + 1.57i)8-s + (−1.58 − 2.54i)9-s + (0.0907 − 0.0447i)10-s + (0.587 − 2.19i)11-s + (0.394 − 3.44i)12-s + (0.104 + 0.388i)13-s + (−1.57 + 1.38i)14-s + (−0.0301 + 0.120i)15-s + (2.82 − 2.82i)16-s − 0.851·17-s + ⋯
 L(s)  = 1 + (−0.980 + 0.195i)2-s + (0.485 − 0.874i)3-s + (0.923 − 0.382i)4-s + (−0.0309 + 0.00828i)5-s + (−0.305 + 0.952i)6-s + (0.484 − 0.279i)7-s + (−0.831 + 0.555i)8-s + (−0.528 − 0.849i)9-s + (0.0287 − 0.0141i)10-s + (0.177 − 0.660i)11-s + (0.113 − 0.993i)12-s + (0.0288 + 0.107i)13-s + (−0.420 + 0.368i)14-s + (−0.00777 + 0.0310i)15-s + (0.706 − 0.707i)16-s − 0.206·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.565 + 0.825i$ 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: $$(2,\ 144,\ (\ :1/2),\ 0.565 + 0.825i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.769559 - 0.405664i$$ $$L(\frac12)$$ $$\approx$$ $$0.769559 - 0.405664i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.38 - 0.276i)T$$
3 $$1 + (-0.841 + 1.51i)T$$
good5 $$1 + (0.0691 - 0.0185i)T + (4.33 - 2.5i)T^{2}$$
7 $$1 + (-1.28 + 0.740i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (-0.587 + 2.19i)T + (-9.52 - 5.5i)T^{2}$$
13 $$1 + (-0.104 - 0.388i)T + (-11.2 + 6.5i)T^{2}$$
17 $$1 + 0.851T + 17T^{2}$$
19 $$1 + (-3.75 + 3.75i)T - 19iT^{2}$$
23 $$1 + (-7.44 - 4.29i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (4.77 + 1.27i)T + (25.1 + 14.5i)T^{2}$$
31 $$1 + (4.50 - 7.79i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-4.13 - 4.13i)T + 37iT^{2}$$
41 $$1 + (2.05 + 1.18i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (0.669 - 2.49i)T + (-37.2 - 21.5i)T^{2}$$
47 $$1 + (-3.42 - 5.92i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-3.95 - 3.95i)T + 53iT^{2}$$
59 $$1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2}$$
61 $$1 + (3.28 + 0.881i)T + (52.8 + 30.5i)T^{2}$$
67 $$1 + (1.73 + 6.47i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + 0.362iT - 71T^{2}$$
73 $$1 - 15.8iT - 73T^{2}$$
79 $$1 + (5.45 + 9.44i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (5.37 + 1.43i)T + (71.8 + 41.5i)T^{2}$$
89 $$1 + 13.1iT - 89T^{2}$$
97 $$1 + (0.627 + 1.08i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$