# Properties

 Label 2-12e2-144.59-c1-0-13 Degree $2$ Conductor $144$ Sign $0.931 + 0.363i$ 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 + (−0.503 + 1.32i)2-s + (1.05 − 1.36i)3-s + (−1.49 − 1.32i)4-s + (−0.746 − 2.78i)5-s + (1.27 + 2.09i)6-s + (1.16 + 2.02i)7-s + (2.50 − 1.30i)8-s + (−0.753 − 2.90i)9-s + (4.05 + 0.415i)10-s + (1.48 − 5.53i)11-s + (−3.40 + 0.636i)12-s + (1.04 + 3.90i)13-s + (−3.25 + 0.525i)14-s + (−4.60 − 1.93i)15-s + (0.462 + 3.97i)16-s + 6.45i·17-s + ⋯
 L(s)  = 1 + (−0.355 + 0.934i)2-s + (0.611 − 0.790i)3-s + (−0.746 − 0.664i)4-s + (−0.333 − 1.24i)5-s + (0.521 + 0.853i)6-s + (0.440 + 0.763i)7-s + (0.887 − 0.461i)8-s + (−0.251 − 0.967i)9-s + (1.28 + 0.131i)10-s + (0.447 − 1.67i)11-s + (−0.982 + 0.183i)12-s + (0.290 + 1.08i)13-s + (−0.870 + 0.140i)14-s + (−1.19 − 0.498i)15-s + (0.115 + 0.993i)16-s + 1.56i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.931 + 0.363i$ Analytic conductor: $$1.14984$$ Root analytic conductor: $$1.07230$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{144} (59, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ 0.931 + 0.363i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.01708 - 0.191446i$$ $$L(\frac12)$$ $$\approx$$ $$1.01708 - 0.191446i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.503 - 1.32i)T$$
3 $$1 + (-1.05 + 1.36i)T$$
good5 $$1 + (0.746 + 2.78i)T + (-4.33 + 2.5i)T^{2}$$
7 $$1 + (-1.16 - 2.02i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (-1.48 + 5.53i)T + (-9.52 - 5.5i)T^{2}$$
13 $$1 + (-1.04 - 3.90i)T + (-11.2 + 6.5i)T^{2}$$
17 $$1 - 6.45iT - 17T^{2}$$
19 $$1 + (1.50 + 1.50i)T + 19iT^{2}$$
23 $$1 + (0.0418 + 0.0241i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (1.36 - 5.08i)T + (-25.1 - 14.5i)T^{2}$$
31 $$1 + (-1.65 - 0.952i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (-0.489 - 0.489i)T + 37iT^{2}$$
41 $$1 + (0.0155 - 0.0269i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-3.80 - 1.01i)T + (37.2 + 21.5i)T^{2}$$
47 $$1 + (0.0913 + 0.158i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (6.62 - 6.62i)T - 53iT^{2}$$
59 $$1 + (-4.15 + 1.11i)T + (51.0 - 29.5i)T^{2}$$
61 $$1 + (-6.39 - 1.71i)T + (52.8 + 30.5i)T^{2}$$
67 $$1 + (0.808 - 0.216i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 - 1.04iT - 71T^{2}$$
73 $$1 - 4.74iT - 73T^{2}$$
79 $$1 + (7.29 - 4.21i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + (-11.9 - 3.20i)T + (71.8 + 41.5i)T^{2}$$
89 $$1 + 2.85T + 89T^{2}$$
97 $$1 + (-3.29 - 5.71i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$