# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 5 \cdot 7$ Sign $0.443 + 0.896i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)3-s + (1.28 − 1.82i)5-s + (1.75 − 1.97i)7-s − 1.00i·9-s + 2.67·11-s + (−1.22 + 1.22i)13-s + (−0.379 − 2.20i)15-s + (4.74 + 4.74i)17-s + 6.01·19-s + (−0.152 − 2.64i)21-s + (0.175 + 0.175i)23-s + (−1.67 − 4.71i)25-s + (−0.707 − 0.707i)27-s + 0.304i·29-s + 7.25i·31-s + ⋯
 L(s)  = 1 + (0.408 − 0.408i)3-s + (0.576 − 0.816i)5-s + (0.665 − 0.746i)7-s − 0.333i·9-s + 0.805·11-s + (−0.340 + 0.340i)13-s + (−0.0979 − 0.568i)15-s + (1.15 + 1.15i)17-s + 1.38·19-s + (−0.0332 − 0.576i)21-s + (0.0366 + 0.0366i)23-s + (−0.334 − 0.942i)25-s + (−0.136 − 0.136i)27-s + 0.0566i·29-s + 1.30i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1680$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.443 + 0.896i$ motivic weight = $$1$$ character : $\chi_{1680} (433, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1680,\ (\ :1/2),\ 0.443 + 0.896i)$$ $$L(1)$$ $$\approx$$ $$2.628468176$$ $$L(\frac12)$$ $$\approx$$ $$2.628468176$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-0.707 + 0.707i)T$$
5 $$1 + (-1.28 + 1.82i)T$$
7 $$1 + (-1.75 + 1.97i)T$$
good11 $$1 - 2.67T + 11T^{2}$$
13 $$1 + (1.22 - 1.22i)T - 13iT^{2}$$
17 $$1 + (-4.74 - 4.74i)T + 17iT^{2}$$
19 $$1 - 6.01T + 19T^{2}$$
23 $$1 + (-0.175 - 0.175i)T + 23iT^{2}$$
29 $$1 - 0.304iT - 29T^{2}$$
31 $$1 - 7.25iT - 31T^{2}$$
37 $$1 + (0.735 - 0.735i)T - 37iT^{2}$$
41 $$1 + 7.05iT - 41T^{2}$$
43 $$1 + (0.304 + 0.304i)T + 43iT^{2}$$
47 $$1 + (-0.556 - 0.556i)T + 47iT^{2}$$
53 $$1 + (4.99 + 4.99i)T + 53iT^{2}$$
59 $$1 + 7.98T + 59T^{2}$$
61 $$1 - 5.53iT - 61T^{2}$$
67 $$1 + (-3.43 + 3.43i)T - 67iT^{2}$$
71 $$1 + 15.3T + 71T^{2}$$
73 $$1 + (10.0 - 10.0i)T - 73iT^{2}$$
79 $$1 + 11.2iT - 79T^{2}$$
83 $$1 + (4.88 - 4.88i)T - 83iT^{2}$$
89 $$1 + 6.91T + 89T^{2}$$
97 $$1 + (-8.84 - 8.84i)T + 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}