# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7$ Sign $-0.987 + 0.157i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.434 + 1.67i)3-s + (−1.21 + 2.10i)5-s + (0.5 + 0.866i)7-s + (−2.62 + 1.45i)9-s + (0.379 + 0.657i)11-s + (−1.11 + 1.92i)13-s + (−4.06 − 1.12i)15-s − 7.04·17-s + 1.37·19-s + (−1.23 + 1.21i)21-s + (3.51 − 6.09i)23-s + (−0.467 − 0.810i)25-s + (−3.58 − 3.76i)27-s + (−0.418 − 0.724i)29-s + (−0.265 + 0.459i)31-s + ⋯
 L(s)  = 1 + (0.250 + 0.967i)3-s + (−0.544 + 0.943i)5-s + (0.188 + 0.327i)7-s + (−0.874 + 0.485i)9-s + (0.114 + 0.198i)11-s + (−0.309 + 0.535i)13-s + (−1.05 − 0.290i)15-s − 1.70·17-s + 0.314·19-s + (−0.269 + 0.265i)21-s + (0.733 − 1.27i)23-s + (−0.0935 − 0.162i)25-s + (−0.689 − 0.724i)27-s + (−0.0776 − 0.134i)29-s + (−0.0476 + 0.0825i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.987 + 0.157i$ motivic weight = $$1$$ character : $\chi_{1008} (337, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1008,\ (\ :1/2),\ -0.987 + 0.157i)$$ $$L(1)$$ $$\approx$$ $$0.9487134374$$ $$L(\frac12)$$ $$\approx$$ $$0.9487134374$$ $$L(\frac{3}{2})$$ 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 + (-0.434 - 1.67i)T$$
7 $$1 + (-0.5 - 0.866i)T$$
good5 $$1 + (1.21 - 2.10i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-0.379 - 0.657i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (1.11 - 1.92i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + 7.04T + 17T^{2}$$
19 $$1 - 1.37T + 19T^{2}$$
23 $$1 + (-3.51 + 6.09i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (0.418 + 0.724i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (0.265 - 0.459i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 4.53T + 37T^{2}$$
41 $$1 + (4.42 - 7.67i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-3.70 - 6.41i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (3.39 + 5.88i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 0.607T + 53T^{2}$$
59 $$1 + (0.581 - 1.00i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (5.85 + 10.1i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (0.152 - 0.264i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 12.6T + 71T^{2}$$
73 $$1 + 10.1T + 73T^{2}$$
79 $$1 + (-7.62 - 13.2i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-8.18 - 14.1i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + 9.63T + 89T^{2}$$
97 $$1 + (-5.46 - 9.46i)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}