# Properties

 Degree 2 Conductor 3 Sign $-1$ Motivic weight 43 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.51e6·2-s + 1.04e10·3-s − 6.49e12·4-s + 1.66e15·5-s + 1.58e16·6-s − 2.14e18·7-s − 2.31e19·8-s + 1.09e20·9-s + 2.52e21·10-s + 1.22e22·11-s − 6.79e22·12-s − 1.24e24·13-s − 3.25e24·14-s + 1.74e25·15-s + 2.20e25·16-s − 3.24e26·17-s + 1.65e26·18-s + 1.94e27·19-s − 1.08e28·20-s − 2.24e28·21-s + 1.86e28·22-s − 1.30e28·23-s − 2.42e29·24-s + 1.64e30·25-s − 1.88e30·26-s + 1.14e30·27-s + 1.39e31·28-s + ⋯
 L(s)  = 1 + 0.511·2-s + 0.577·3-s − 0.738·4-s + 1.56·5-s + 0.295·6-s − 1.45·7-s − 0.888·8-s + 0.333·9-s + 0.799·10-s + 0.500·11-s − 0.426·12-s − 1.39·13-s − 0.742·14-s + 0.903·15-s + 0.284·16-s − 1.13·17-s + 0.170·18-s + 0.625·19-s − 1.15·20-s − 0.838·21-s + 0.255·22-s − 0.0690·23-s − 0.513·24-s + 1.44·25-s − 0.713·26-s + 0.192·27-s + 1.07·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $-1$ motivic weight = $$43$$ character : $\chi_{3} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 3,\ (\ :43/2),\ -1)$$ $$L(22)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{45}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 - 1.04e10T$$
good2 $$1 - 1.51e6T + 8.79e12T^{2}$$
5 $$1 - 1.66e15T + 1.13e30T^{2}$$
7 $$1 + 2.14e18T + 2.18e36T^{2}$$
11 $$1 - 1.22e22T + 6.02e44T^{2}$$
13 $$1 + 1.24e24T + 7.93e47T^{2}$$
17 $$1 + 3.24e26T + 8.11e52T^{2}$$
19 $$1 - 1.94e27T + 9.69e54T^{2}$$
23 $$1 + 1.30e28T + 3.58e58T^{2}$$
29 $$1 + 3.97e31T + 7.64e62T^{2}$$
31 $$1 + 2.09e32T + 1.34e64T^{2}$$
37 $$1 - 2.50e33T + 2.70e67T^{2}$$
41 $$1 + 4.85e34T + 2.23e69T^{2}$$
43 $$1 + 1.71e35T + 1.73e70T^{2}$$
47 $$1 + 5.66e35T + 7.94e71T^{2}$$
53 $$1 + 3.68e36T + 1.39e74T^{2}$$
59 $$1 + 2.83e37T + 1.40e76T^{2}$$
61 $$1 - 1.88e38T + 5.87e76T^{2}$$
67 $$1 - 4.53e37T + 3.32e78T^{2}$$
71 $$1 - 9.40e39T + 4.01e79T^{2}$$
73 $$1 - 1.85e40T + 1.32e80T^{2}$$
79 $$1 - 2.01e40T + 3.96e81T^{2}$$
83 $$1 - 5.04e40T + 3.31e82T^{2}$$
89 $$1 - 2.49e41T + 6.66e83T^{2}$$
97 $$1 - 5.50e42T + 2.69e85T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}