# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $0.724 - 0.689i$ Motivic weight 15 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (166. + 71.1i)2-s + (8.61 − 3.78e3i)3-s + (2.26e4 + 2.36e4i)4-s + 9.97e4i·5-s + (2.71e5 − 6.29e5i)6-s + 1.78e6i·7-s + (2.08e6 + 5.55e6i)8-s + (−1.43e7 − 6.52e4i)9-s + (−7.10e6 + 1.66e7i)10-s + 1.16e8·11-s + (8.99e7 − 8.55e7i)12-s + 2.21e8·13-s + (−1.26e8 + 2.96e8i)14-s + (3.77e8 + 8.59e5i)15-s + (−4.88e7 + 1.07e9i)16-s − 1.04e9i·17-s + ⋯
 L(s)  = 1 + (0.919 + 0.393i)2-s + (0.00227 − 0.999i)3-s + (0.690 + 0.723i)4-s + 0.571i·5-s + (0.395 − 0.918i)6-s + 0.818i·7-s + (0.350 + 0.936i)8-s + (−0.999 − 0.00454i)9-s + (−0.224 + 0.525i)10-s + 1.79·11-s + (0.724 − 0.689i)12-s + 0.978·13-s + (−0.321 + 0.752i)14-s + (0.571 + 0.00129i)15-s + (−0.0455 + 0.998i)16-s − 0.619i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(16-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $0.724 - 0.689i$ motivic weight = $$15$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :15/2),\ 0.724 - 0.689i)$ $L(8)$ $\approx$ $3.22377 + 1.28827i$ $L(\frac12)$ $\approx$ $3.22377 + 1.28827i$ $L(\frac{17}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ 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 + (-166. - 71.1i)T$$
3 $$1 + (-8.61 + 3.78e3i)T$$
good5 $$1 - 9.97e4iT - 3.05e10T^{2}$$
7 $$1 - 1.78e6iT - 4.74e12T^{2}$$
11 $$1 - 1.16e8T + 4.17e15T^{2}$$
13 $$1 - 2.21e8T + 5.11e16T^{2}$$
17 $$1 + 1.04e9iT - 2.86e18T^{2}$$
19 $$1 - 4.40e9iT - 1.51e19T^{2}$$
23 $$1 + 1.18e10T + 2.66e20T^{2}$$
29 $$1 - 3.11e10iT - 8.62e21T^{2}$$
31 $$1 + 8.47e10iT - 2.34e22T^{2}$$
37 $$1 + 5.63e11T + 3.33e23T^{2}$$
41 $$1 + 1.67e12iT - 1.55e24T^{2}$$
43 $$1 - 6.70e11iT - 3.17e24T^{2}$$
47 $$1 + 4.09e12T + 1.20e25T^{2}$$
53 $$1 + 1.54e13iT - 7.31e25T^{2}$$
59 $$1 - 1.68e13T + 3.65e26T^{2}$$
61 $$1 + 2.24e13T + 6.02e26T^{2}$$
67 $$1 + 8.16e11iT - 2.46e27T^{2}$$
71 $$1 + 2.78e13T + 5.87e27T^{2}$$
73 $$1 + 9.23e13T + 8.90e27T^{2}$$
79 $$1 - 1.17e14iT - 2.91e28T^{2}$$
83 $$1 - 3.73e14T + 6.11e28T^{2}$$
89 $$1 - 9.06e13iT - 1.74e29T^{2}$$
97 $$1 - 2.89e14T + 6.33e29T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}