# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $-0.985 - 0.170i$ Motivic weight 15 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (80.5 − 162. i)2-s + (2.76e3 − 2.58e3i)3-s + (−1.97e4 − 2.61e4i)4-s − 3.32e5i·5-s + (−1.96e5 − 6.57e5i)6-s + 2.13e6i·7-s + (−5.82e6 + 1.09e6i)8-s + (9.67e5 − 1.43e7i)9-s + (−5.39e7 − 2.68e7i)10-s + 7.24e7·11-s + (−1.22e8 − 2.11e7i)12-s + 5.97e7·13-s + (3.46e8 + 1.72e8i)14-s + (−8.60e8 − 9.20e8i)15-s + (−2.91e8 + 1.03e9i)16-s + 9.88e8i·17-s + ⋯
 L(s)  = 1 + (0.445 − 0.895i)2-s + (0.730 − 0.682i)3-s + (−0.603 − 0.797i)4-s − 1.90i·5-s + (−0.286 − 0.958i)6-s + 0.979i·7-s + (−0.982 + 0.185i)8-s + (0.0674 − 0.997i)9-s + (−1.70 − 0.847i)10-s + 1.12·11-s + (−0.985 − 0.170i)12-s + 0.264·13-s + (0.877 + 0.436i)14-s + (−1.30 − 1.39i)15-s + (−0.271 + 0.962i)16-s + 0.584i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(16-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $-0.985 - 0.170i$ motivic weight = $$15$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :15/2),\ -0.985 - 0.170i)$ $L(8)$ $\approx$ $0.228375 + 2.66086i$ $L(\frac12)$ $\approx$ $0.228375 + 2.66086i$ $L(\frac{17}{2})$ 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 + (-80.5 + 162. i)T$$
3 $$1 + (-2.76e3 + 2.58e3i)T$$
good5 $$1 + 3.32e5iT - 3.05e10T^{2}$$
7 $$1 - 2.13e6iT - 4.74e12T^{2}$$
11 $$1 - 7.24e7T + 4.17e15T^{2}$$
13 $$1 - 5.97e7T + 5.11e16T^{2}$$
17 $$1 - 9.88e8iT - 2.86e18T^{2}$$
19 $$1 + 6.22e7iT - 1.51e19T^{2}$$
23 $$1 - 2.88e9T + 2.66e20T^{2}$$
29 $$1 - 3.09e10iT - 8.62e21T^{2}$$
31 $$1 + 4.97e10iT - 2.34e22T^{2}$$
37 $$1 - 2.95e11T + 3.33e23T^{2}$$
41 $$1 + 1.49e12iT - 1.55e24T^{2}$$
43 $$1 + 2.69e12iT - 3.17e24T^{2}$$
47 $$1 - 4.45e12T + 1.20e25T^{2}$$
53 $$1 - 3.63e12iT - 7.31e25T^{2}$$
59 $$1 + 2.17e13T + 3.65e26T^{2}$$
61 $$1 + 3.26e12T + 6.02e26T^{2}$$
67 $$1 - 7.11e13iT - 2.46e27T^{2}$$
71 $$1 - 4.18e13T + 5.87e27T^{2}$$
73 $$1 - 1.11e14T + 8.90e27T^{2}$$
79 $$1 + 1.98e14iT - 2.91e28T^{2}$$
83 $$1 + 1.29e14T + 6.11e28T^{2}$$
89 $$1 - 1.60e14iT - 1.74e29T^{2}$$
97 $$1 + 1.03e15T + 6.33e29T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}