# Properties

 Degree 2 Conductor $2^{5} \cdot 3 \cdot 7$ Sign $0.762 - 0.647i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.46 + 0.916i)3-s + 1.83·5-s + i·7-s + (1.31 + 2.69i)9-s − 5.57i·13-s + (2.69 + 1.68i)15-s + 6.08i·17-s + 6.93·19-s + (−0.916 + 1.46i)21-s + 0.697·23-s − 1.63·25-s + (−0.530 + 5.16i)27-s − 7.11·29-s + 1.83i·35-s − 1.87i·37-s + ⋯
 L(s)  = 1 + (0.848 + 0.529i)3-s + 0.819·5-s + 0.377i·7-s + (0.439 + 0.898i)9-s − 1.54i·13-s + (0.695 + 0.433i)15-s + 1.47i·17-s + 1.59·19-s + (−0.200 + 0.320i)21-s + 0.145·23-s − 0.327·25-s + (−0.102 + 0.994i)27-s − 1.32·29-s + 0.309i·35-s − 0.308i·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$672$$    =    $$2^{5} \cdot 3 \cdot 7$$ $$\varepsilon$$ = $0.762 - 0.647i$ motivic weight = $$1$$ character : $\chi_{672} (239, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 672,\ (\ :1/2),\ 0.762 - 0.647i)$ $L(1)$ $\approx$ $2.16420 + 0.794684i$ $L(\frac12)$ $\approx$ $2.16420 + 0.794684i$ $L(\frac{3}{2})$ 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 + (-1.46 - 0.916i)T$$
7 $$1 - iT$$
good5 $$1 - 1.83T + 5T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + 5.57iT - 13T^{2}$$
17 $$1 - 6.08iT - 17T^{2}$$
19 $$1 - 6.93T + 19T^{2}$$
23 $$1 - 0.697T + 23T^{2}$$
29 $$1 + 7.11T + 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + 1.87iT - 37T^{2}$$
41 $$1 + 4.69iT - 41T^{2}$$
43 $$1 + 1.36T + 43T^{2}$$
47 $$1 + 3.44T + 47T^{2}$$
53 $$1 - 12.1T + 53T^{2}$$
59 $$1 + 8.94iT - 59T^{2}$$
61 $$1 - 4.30iT - 61T^{2}$$
67 $$1 + 4.51T + 67T^{2}$$
71 $$1 + 6.08T + 71T^{2}$$
73 $$1 + 6T + 73T^{2}$$
79 $$1 - 11.7iT - 79T^{2}$$
83 $$1 + 0.438iT - 83T^{2}$$
89 $$1 + 2.64iT - 89T^{2}$$
97 $$1 - 11.0T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}