# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $-0.964 + 0.262i$ Motivic weight 19 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (627. + 362. i)2-s + (−8.68e3 + 3.29e4i)3-s + (2.61e5 + 4.54e5i)4-s + 6.03e6i·5-s + (−1.73e7 + 1.75e7i)6-s + 9.28e7i·7-s + (−2.07e5 + 3.79e8i)8-s + (−1.01e9 − 5.72e8i)9-s + (−2.18e9 + 3.78e9i)10-s + 6.13e9·11-s + (−1.72e10 + 4.69e9i)12-s + 4.99e10·13-s + (−3.36e10 + 5.82e10i)14-s + (−1.98e11 − 5.23e10i)15-s + (−1.37e11 + 2.37e11i)16-s − 2.01e11i·17-s + ⋯
 L(s)  = 1 + (0.865 + 0.500i)2-s + (−0.254 + 0.967i)3-s + (0.499 + 0.866i)4-s + 1.38i·5-s + (−0.704 + 0.709i)6-s + 0.870i·7-s + (−0.000545 + 0.999i)8-s + (−0.870 − 0.492i)9-s + (−0.690 + 1.19i)10-s + 0.784·11-s + (−0.964 + 0.262i)12-s + 1.30·13-s + (−0.435 + 0.753i)14-s + (−1.33 − 0.351i)15-s + (−0.500 + 0.865i)16-s − 0.412i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(20-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $-0.964 + 0.262i$ motivic weight = $$19$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :19/2),\ -0.964 + 0.262i)$ $L(10)$ $\approx$ $3.119837508$ $L(\frac12)$ $\approx$ $3.119837508$ $L(\frac{21}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ 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 + (-627. - 362. i)T$$
3 $$1 + (8.68e3 - 3.29e4i)T$$
good5 $$1 - 6.03e6iT - 1.90e13T^{2}$$
7 $$1 - 9.28e7iT - 1.13e16T^{2}$$
11 $$1 - 6.13e9T + 6.11e19T^{2}$$
13 $$1 - 4.99e10T + 1.46e21T^{2}$$
17 $$1 + 2.01e11iT - 2.39e23T^{2}$$
19 $$1 + 4.67e11iT - 1.97e24T^{2}$$
23 $$1 - 8.46e12T + 7.46e25T^{2}$$
29 $$1 + 2.39e13iT - 6.10e27T^{2}$$
31 $$1 + 1.41e14iT - 2.16e28T^{2}$$
37 $$1 + 1.37e15T + 6.24e29T^{2}$$
41 $$1 - 3.22e15iT - 4.39e30T^{2}$$
43 $$1 + 5.84e15iT - 1.08e31T^{2}$$
47 $$1 - 1.11e16T + 5.88e31T^{2}$$
53 $$1 - 3.17e16iT - 5.77e32T^{2}$$
59 $$1 + 1.59e16T + 4.42e33T^{2}$$
61 $$1 - 1.40e17T + 8.34e33T^{2}$$
67 $$1 - 3.36e16iT - 4.95e34T^{2}$$
71 $$1 - 6.45e17T + 1.49e35T^{2}$$
73 $$1 + 6.40e17T + 2.53e35T^{2}$$
79 $$1 + 3.85e17iT - 1.13e36T^{2}$$
83 $$1 + 5.19e17T + 2.90e36T^{2}$$
89 $$1 + 5.87e17iT - 1.09e37T^{2}$$
97 $$1 + 4.81e18T + 5.60e37T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}