# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{3} \cdot 7$ Sign $0.888 + 0.458i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 − 2.59i)5-s + (2 + 1.73i)7-s + (1.5 − 2.59i)11-s + (−2.5 + 4.33i)13-s + (1.5 + 2.59i)17-s + (2.5 − 4.33i)19-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s + (−1.5 − 2.59i)29-s + 4·31-s + (1.5 − 7.79i)35-s + (3.5 − 6.06i)37-s + (−4.5 + 7.79i)41-s + (5.5 + 9.52i)43-s + (1.00 + 6.92i)49-s + ⋯
 L(s)  = 1 + (−0.670 − 1.16i)5-s + (0.755 + 0.654i)7-s + (0.452 − 0.783i)11-s + (−0.693 + 1.20i)13-s + (0.363 + 0.630i)17-s + (0.573 − 0.993i)19-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.278 − 0.482i)29-s + 0.718·31-s + (0.253 − 1.31i)35-s + (0.575 − 0.996i)37-s + (−0.702 + 1.21i)41-s + (0.838 + 1.45i)43-s + (0.142 + 0.989i)49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.888 + 0.458i$ motivic weight = $$1$$ character : $\chi_{3024} (2305, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3024,\ (\ :1/2),\ 0.888 + 0.458i)$$ $$L(1)$$ $$\approx$$ $$1.785916447$$ $$L(\frac12)$$ $$\approx$$ $$1.785916447$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + (-2 - 1.73i)T$$
good5 $$1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 - 4T + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}