Properties

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

Related objects

Dirichlet series

 L(s)  = 1 + 4.46i·5-s − 2.64·7-s + 4.46i·11-s + 7.34i·17-s + 8.64·19-s + 2.88i·23-s − 14.9·25-s + 8.29·31-s − 11.8i·35-s − 3.93·37-s + 2.88i·41-s + 7.00·49-s − 19.9·55-s − 10.2i·71-s − 11.8i·77-s + ⋯
 L(s)  = 1 + 1.99i·5-s − 0.999·7-s + 1.34i·11-s + 1.78i·17-s + 1.98·19-s + 0.601i·23-s − 2.98·25-s + 1.48·31-s − 1.99i·35-s − 0.647·37-s + 0.450i·41-s + 49-s − 2.68·55-s − 1.21i·71-s − 1.34i·77-s + ⋯

Functional equation

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

Invariants

 $$d$$ = $$2$$ $$N$$ = $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{3024} (1567, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3024,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$\approx$$ $$1.370620880$$ $$L(\frac12)$$ $$\approx$$ $$1.370620880$$ $$L(\frac{3}{2})$$ 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 + 2.64T$$
good5 $$1 - 4.46iT - 5T^{2}$$
11 $$1 - 4.46iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 7.34iT - 17T^{2}$$
19 $$1 - 8.64T + 19T^{2}$$
23 $$1 - 2.88iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 8.29T + 31T^{2}$$
37 $$1 + 3.93T + 37T^{2}$$
41 $$1 - 2.88iT - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 10.2iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 14.9iT - 89T^{2}$$
97 $$1 - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}