# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $0.888 - 0.459i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.73 + 2.23i)2-s + (−3.46 − 3.87i)3-s + (−2.00 + 7.74i)4-s − 8.94i·5-s + (2.66 − 14.4i)6-s + 7.74i·7-s + (−20.7 + 8.94i)8-s + (−3.00 + 26.8i)9-s + (20.0 − 15.4i)10-s + 34.6·11-s + (36.9 − 19.0i)12-s − 10·13-s + (−17.3 + 13.4i)14-s + (−34.6 + 30.9i)15-s + (−56 − 30.9i)16-s − 35.7i·17-s + ⋯
 L(s)  = 1 + (0.612 + 0.790i)2-s + (−0.666 − 0.745i)3-s + (−0.250 + 0.968i)4-s − 0.799i·5-s + (0.181 − 0.983i)6-s + 0.418i·7-s + (−0.918 + 0.395i)8-s + (−0.111 + 0.993i)9-s + (0.632 − 0.489i)10-s + 0.949·11-s + (0.888 − 0.459i)12-s − 0.213·13-s + (−0.330 + 0.256i)14-s + (−0.596 + 0.533i)15-s + (−0.875 − 0.484i)16-s − 0.510i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $0.888 - 0.459i$ motivic weight = $$3$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :3/2),\ 0.888 - 0.459i)$ $L(2)$ $\approx$ $0.972332 + 0.236425i$ $L(\frac12)$ $\approx$ $0.972332 + 0.236425i$ $L(\frac{5}{2})$ 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 + (-1.73 - 2.23i)T$$
3 $$1 + (3.46 + 3.87i)T$$
good5 $$1 + 8.94iT - 125T^{2}$$
7 $$1 - 7.74iT - 343T^{2}$$
11 $$1 - 34.6T + 1.33e3T^{2}$$
13 $$1 + 10T + 2.19e3T^{2}$$
17 $$1 + 35.7iT - 4.91e3T^{2}$$
19 $$1 + 69.7iT - 6.85e3T^{2}$$
23 $$1 + 96.9T + 1.21e4T^{2}$$
29 $$1 - 152. iT - 2.43e4T^{2}$$
31 $$1 - 224. iT - 2.97e4T^{2}$$
37 $$1 + 130T + 5.06e4T^{2}$$
41 $$1 - 125. iT - 6.89e4T^{2}$$
43 $$1 + 224. iT - 7.95e4T^{2}$$
47 $$1 + 193.T + 1.03e5T^{2}$$
53 $$1 + 545. iT - 1.48e5T^{2}$$
59 $$1 - 173.T + 2.05e5T^{2}$$
61 $$1 + 442T + 2.26e5T^{2}$$
67 $$1 - 735. iT - 3.00e5T^{2}$$
71 $$1 - 1.03e3T + 3.57e5T^{2}$$
73 $$1 - 410T + 3.89e5T^{2}$$
79 $$1 + 85.2iT - 4.93e5T^{2}$$
83 $$1 + 1.25e3T + 5.71e5T^{2}$$
89 $$1 + 840. iT - 7.04e5T^{2}$$
97 $$1 - 770T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}