# Properties

 Degree 2 Conductor $2 \cdot 3$ Sign $-0.974 + 0.225i$ Motivic weight 16 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 181. i·2-s + (1.47e3 + 6.39e3i)3-s − 3.27e4·4-s + 4.41e5i·5-s + (−1.15e6 + 2.67e5i)6-s + 2.81e6·7-s − 5.93e6i·8-s + (−3.86e7 + 1.89e7i)9-s − 7.99e7·10-s − 1.92e8i·11-s + (−4.84e7 − 2.09e8i)12-s − 1.24e9·13-s + 5.09e8i·14-s + (−2.82e9 + 6.52e8i)15-s + 1.07e9·16-s + 1.41e9i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (0.225 + 0.974i)3-s − 0.500·4-s + 1.13i·5-s + (−0.688 + 0.159i)6-s + 0.488·7-s − 0.353i·8-s + (−0.898 + 0.439i)9-s − 0.799·10-s − 0.900i·11-s + (−0.112 − 0.487i)12-s − 1.52·13-s + 0.345i·14-s + (−1.10 + 0.254i)15-s + 0.250·16-s + 0.203i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(17-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6$$    =    $$2 \cdot 3$$ $$\varepsilon$$ = $-0.974 + 0.225i$ motivic weight = $$16$$ character : $\chi_{6} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6,\ (\ :8),\ -0.974 + 0.225i)$ $L(\frac{17}{2})$ $\approx$ $0.156209 - 1.36800i$ $L(\frac12)$ $\approx$ $0.156209 - 1.36800i$ $L(9)$ 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 - 181. iT$$
3 $$1 + (-1.47e3 - 6.39e3i)T$$
good5 $$1 - 4.41e5iT - 1.52e11T^{2}$$
7 $$1 - 2.81e6T + 3.32e13T^{2}$$
11 $$1 + 1.92e8iT - 4.59e16T^{2}$$
13 $$1 + 1.24e9T + 6.65e17T^{2}$$
17 $$1 - 1.41e9iT - 4.86e19T^{2}$$
19 $$1 - 2.68e10T + 2.88e20T^{2}$$
23 $$1 - 1.20e11iT - 6.13e21T^{2}$$
29 $$1 - 6.43e11iT - 2.50e23T^{2}$$
31 $$1 + 1.14e12T + 7.27e23T^{2}$$
37 $$1 - 4.25e12T + 1.23e25T^{2}$$
41 $$1 - 3.12e12iT - 6.37e25T^{2}$$
43 $$1 - 4.47e12T + 1.36e26T^{2}$$
47 $$1 - 1.17e13iT - 5.66e26T^{2}$$
53 $$1 + 4.41e13iT - 3.87e27T^{2}$$
59 $$1 - 1.42e14iT - 2.15e28T^{2}$$
61 $$1 - 2.29e13T + 3.67e28T^{2}$$
67 $$1 + 1.94e14T + 1.64e29T^{2}$$
71 $$1 - 3.85e13iT - 4.16e29T^{2}$$
73 $$1 - 1.42e15T + 6.50e29T^{2}$$
79 $$1 - 1.03e15T + 2.30e30T^{2}$$
83 $$1 - 1.75e15iT - 5.07e30T^{2}$$
89 $$1 - 1.35e15iT - 1.54e31T^{2}$$
97 $$1 - 5.20e15T + 6.14e31T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}