# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2}$ Sign $-0.737 + 0.675i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.366 + 1.36i)2-s + (−1.5 − 0.866i)3-s + (−1.73 + i)4-s + (−3.73 − i)5-s + (0.633 − 2.36i)6-s + (−0.633 − 0.366i)7-s + (−2 − 1.99i)8-s + (1.5 + 2.59i)9-s − 5.46i·10-s + (−0.767 − 2.86i)11-s + 3.46·12-s + (−1.63 + 6.09i)13-s + (0.267 − i)14-s + (4.73 + 4.73i)15-s + (1.99 − 3.46i)16-s − 2.26·17-s + ⋯
 L(s)  = 1 + (0.258 + 0.965i)2-s + (−0.866 − 0.499i)3-s + (−0.866 + 0.5i)4-s + (−1.66 − 0.447i)5-s + (0.258 − 0.965i)6-s + (−0.239 − 0.138i)7-s + (−0.707 − 0.707i)8-s + (0.5 + 0.866i)9-s − 1.72i·10-s + (−0.231 − 0.864i)11-s + 0.999·12-s + (−0.453 + 1.69i)13-s + (0.0716 − 0.267i)14-s + (1.22 + 1.22i)15-s + (0.499 − 0.866i)16-s − 0.550·17-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

 $$d$$ = $$2$$ $$N$$ = $$144$$    =    $$2^{4} \cdot 3^{2}$$ $$\varepsilon$$ = $-0.737 + 0.675i$ motivic weight = $$1$$ character : $\chi_{144} (61, \cdot )$ primitive : yes self-dual : no analytic rank = $$1$$ Selberg data = $$(2,\ 144,\ (\ :1/2),\ -0.737 + 0.675i)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.366 - 1.36i)T$$
3 $$1 + (1.5 + 0.866i)T$$
good5 $$1 + (3.73 + i)T + (4.33 + 2.5i)T^{2}$$
7 $$1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (0.767 + 2.86i)T + (-9.52 + 5.5i)T^{2}$$
13 $$1 + (1.63 - 6.09i)T + (-11.2 - 6.5i)T^{2}$$
17 $$1 + 2.26T + 17T^{2}$$
19 $$1 + (0.633 + 0.633i)T + 19iT^{2}$$
23 $$1 + (1.09 - 0.633i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (2.36 - 0.633i)T + (25.1 - 14.5i)T^{2}$$
31 $$1 + (3.73 + 6.46i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-1.26 + 1.26i)T - 37iT^{2}$$
41 $$1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.330 - 1.23i)T + (-37.2 + 21.5i)T^{2}$$
47 $$1 + (4.83 - 8.36i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (0.535 - 0.535i)T - 53iT^{2}$$
59 $$1 + (-4.96 - 1.33i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (3 - 0.803i)T + (52.8 - 30.5i)T^{2}$$
67 $$1 + (-1.40 + 5.23i)T + (-58.0 - 33.5i)T^{2}$$
71 $$1 + 10.9iT - 71T^{2}$$
73 $$1 + 9.73iT - 73T^{2}$$
79 $$1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (1.36 - 0.366i)T + (71.8 - 41.5i)T^{2}$$
89 $$1 + 2iT - 89T^{2}$$
97 $$1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}