# Properties

 Degree $2$ Conductor $144$ Sign $0.945 + 0.324i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.32 − 0.505i)2-s + (1.48 − 1.33i)4-s + (2.10 + 2.10i)5-s − 4.40·7-s + (1.28 − 2.51i)8-s + (3.84 + 1.71i)10-s + (−0.215 + 0.215i)11-s + (−2.73 − 2.73i)13-s + (−5.82 + 2.22i)14-s + (0.430 − 3.97i)16-s + 2.36i·17-s + (0.758 − 0.758i)19-s + (5.94 + 0.320i)20-s + (−0.175 + 0.393i)22-s − 1.75i·23-s + ⋯
 L(s)  = 1 + (0.933 − 0.357i)2-s + (0.744 − 0.667i)4-s + (0.941 + 0.941i)5-s − 1.66·7-s + (0.456 − 0.889i)8-s + (1.21 + 0.542i)10-s + (−0.0650 + 0.0650i)11-s + (−0.758 − 0.758i)13-s + (−1.55 + 0.595i)14-s + (0.107 − 0.994i)16-s + 0.573i·17-s + (0.174 − 0.174i)19-s + (1.32 + 0.0717i)20-s + (−0.0374 + 0.0839i)22-s − 0.366i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.945 + 0.324i$ Motivic weight: $$1$$ Character: $\chi_{144} (35, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ 0.945 + 0.324i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.76228 - 0.294013i$$ $$L(\frac12)$$ $$\approx$$ $$1.76228 - 0.294013i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.32 + 0.505i)T$$
3 $$1$$
good5 $$1 + (-2.10 - 2.10i)T + 5iT^{2}$$
7 $$1 + 4.40T + 7T^{2}$$
11 $$1 + (0.215 - 0.215i)T - 11iT^{2}$$
13 $$1 + (2.73 + 2.73i)T + 13iT^{2}$$
17 $$1 - 2.36iT - 17T^{2}$$
19 $$1 + (-0.758 + 0.758i)T - 19iT^{2}$$
23 $$1 + 1.75iT - 23T^{2}$$
29 $$1 + (5.54 - 5.54i)T - 29iT^{2}$$
31 $$1 - 9.01iT - 31T^{2}$$
37 $$1 + (-3.10 + 3.10i)T - 37iT^{2}$$
41 $$1 - 10.1T + 41T^{2}$$
43 $$1 + (3.54 + 3.54i)T + 43iT^{2}$$
47 $$1 - 3.90T + 47T^{2}$$
53 $$1 + (2.71 + 2.71i)T + 53iT^{2}$$
59 $$1 + (-3.40 + 3.40i)T - 59iT^{2}$$
61 $$1 + (1.75 + 1.75i)T + 61iT^{2}$$
67 $$1 + (-9.11 + 9.11i)T - 67iT^{2}$$
71 $$1 - 11.8iT - 71T^{2}$$
73 $$1 + 0.482iT - 73T^{2}$$
79 $$1 + 6.88iT - 79T^{2}$$
83 $$1 + (-4.79 - 4.79i)T + 83iT^{2}$$
89 $$1 + 7.00T + 89T^{2}$$
97 $$1 + 3.34T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$