# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $-0.354 + 0.934i$ Motivic weight 19 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (641. − 336. i)2-s + (−3.30e4 + 8.17e3i)3-s + (2.98e5 − 4.31e5i)4-s + 2.18e6i·5-s + (−1.84e7 + 1.63e7i)6-s + 4.59e7i·7-s + (4.61e7 − 3.76e8i)8-s + (1.02e9 − 5.41e8i)9-s + (7.34e8 + 1.40e9i)10-s + 1.63e8·11-s + (−6.34e9 + 1.67e10i)12-s + 2.15e9·13-s + (1.54e10 + 2.94e10i)14-s + (−1.78e10 − 7.22e10i)15-s + (−9.71e10 − 2.57e11i)16-s − 3.11e11i·17-s + ⋯
 L(s)  = 1 + (0.885 − 0.464i)2-s + (−0.970 + 0.239i)3-s + (0.568 − 0.822i)4-s + 0.500i·5-s + (−0.748 + 0.663i)6-s + 0.430i·7-s + (0.121 − 0.992i)8-s + (0.885 − 0.465i)9-s + (0.232 + 0.442i)10-s + 0.0208·11-s + (−0.354 + 0.934i)12-s + 0.0563·13-s + (0.200 + 0.381i)14-s + (−0.119 − 0.485i)15-s + (−0.353 − 0.935i)16-s − 0.637i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(20-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $-0.354 + 0.934i$ motivic weight = $$19$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :19/2),\ -0.354 + 0.934i)$ $L(10)$ $\approx$ $1.895970983$ $L(\frac12)$ $\approx$ $1.895970983$ $L(\frac{21}{2})$ 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 + (-641. + 336. i)T$$
3 $$1 + (3.30e4 - 8.17e3i)T$$
good5 $$1 - 2.18e6iT - 1.90e13T^{2}$$
7 $$1 - 4.59e7iT - 1.13e16T^{2}$$
11 $$1 - 1.63e8T + 6.11e19T^{2}$$
13 $$1 - 2.15e9T + 1.46e21T^{2}$$
17 $$1 + 3.11e11iT - 2.39e23T^{2}$$
19 $$1 + 2.29e12iT - 1.97e24T^{2}$$
23 $$1 + 1.23e13T + 7.46e25T^{2}$$
29 $$1 + 1.30e14iT - 6.10e27T^{2}$$
31 $$1 + 8.17e13iT - 2.16e28T^{2}$$
37 $$1 - 2.92e14T + 6.24e29T^{2}$$
41 $$1 + 2.15e15iT - 4.39e30T^{2}$$
43 $$1 - 6.38e14iT - 1.08e31T^{2}$$
47 $$1 + 5.11e15T + 5.88e31T^{2}$$
53 $$1 - 5.34e15iT - 5.77e32T^{2}$$
59 $$1 - 5.99e16T + 4.42e33T^{2}$$
61 $$1 - 8.58e16T + 8.34e33T^{2}$$
67 $$1 - 3.62e17iT - 4.95e34T^{2}$$
71 $$1 + 4.95e17T + 1.49e35T^{2}$$
73 $$1 - 7.06e17T + 2.53e35T^{2}$$
79 $$1 + 1.01e18iT - 1.13e36T^{2}$$
83 $$1 - 4.60e17T + 2.90e36T^{2}$$
89 $$1 - 5.30e18iT - 1.09e37T^{2}$$
97 $$1 + 1.13e19T + 5.60e37T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}