# Properties

 Degree 2 Conductor $2^{3} \cdot 3$ Sign $0.774 - 0.632i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.36 + 1.46i)2-s − 1.73·3-s + (−0.267 + 3.99i)4-s − 7.98i·5-s + (−2.36 − 2.53i)6-s + 2.13i·7-s + (−6.19 + 5.06i)8-s + 2.99·9-s + (11.6 − 10.9i)10-s − 8·11-s + (0.464 − 6.91i)12-s + 11.6i·13-s + (−3.12 + 2.92i)14-s + 13.8i·15-s + (−15.8 − 2.13i)16-s + 11.8·17-s + ⋯
 L(s)  = 1 + (0.683 + 0.730i)2-s − 0.577·3-s + (−0.0669 + 0.997i)4-s − 1.59i·5-s + (−0.394 − 0.421i)6-s + 0.305i·7-s + (−0.774 + 0.632i)8-s + 0.333·9-s + (1.16 − 1.09i)10-s − 0.727·11-s + (0.0386 − 0.576i)12-s + 0.898i·13-s + (−0.223 + 0.208i)14-s + 0.921i·15-s + (−0.991 − 0.133i)16-s + 0.697·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$24$$    =    $$2^{3} \cdot 3$$ $$\varepsilon$$ = $0.774 - 0.632i$ motivic weight = $$2$$ character : $\chi_{24} (19, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 24,\ (\ :1),\ 0.774 - 0.632i)$ $L(\frac{3}{2})$ $\approx$ $0.985254 + 0.351206i$ $L(\frac12)$ $\approx$ $0.985254 + 0.351206i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (-1.36 - 1.46i)T$$
3 $$1 + 1.73T$$
good5 $$1 + 7.98iT - 25T^{2}$$
7 $$1 - 2.13iT - 49T^{2}$$
11 $$1 + 8T + 121T^{2}$$
13 $$1 - 11.6iT - 169T^{2}$$
17 $$1 - 11.8T + 289T^{2}$$
19 $$1 - 14.9T + 361T^{2}$$
23 $$1 + 4.27iT - 529T^{2}$$
29 $$1 - 0.573iT - 841T^{2}$$
31 $$1 + 57.4iT - 961T^{2}$$
37 $$1 - 27.6iT - 1.36e3T^{2}$$
41 $$1 + 31.5T + 1.68e3T^{2}$$
43 $$1 - 28.7T + 1.84e3T^{2}$$
47 $$1 - 59.5iT - 2.20e3T^{2}$$
53 $$1 - 31.3iT - 2.80e3T^{2}$$
59 $$1 + 52.7T + 3.48e3T^{2}$$
61 $$1 + 59.5iT - 3.72e3T^{2}$$
67 $$1 + 84.7T + 4.48e3T^{2}$$
71 $$1 + 42.4iT - 5.04e3T^{2}$$
73 $$1 + 5.42T + 5.32e3T^{2}$$
79 $$1 - 44.6iT - 6.24e3T^{2}$$
83 $$1 - 67.7T + 6.88e3T^{2}$$
89 $$1 + 133.T + 7.92e3T^{2}$$
97 $$1 - 97.1T + 9.40e3T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}