# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.70 + 0.300i)3-s + (−1.26 + 2.19i)5-s + (0.5 + 0.866i)7-s + (2.81 − 1.02i)9-s + (0.233 + 0.405i)11-s + (−2.91 + 5.04i)13-s + (1.5 − 4.12i)15-s + 3.87·17-s + 2.18·19-s + (−1.11 − 1.32i)21-s + (−0.0530 + 0.0918i)23-s + (−0.705 − 1.22i)25-s + (−4.49 + 2.59i)27-s + (−4.39 − 7.60i)29-s + (−3.84 + 6.65i)31-s + ⋯
 L(s)  = 1 + (−0.984 + 0.173i)3-s + (−0.566 + 0.980i)5-s + (0.188 + 0.327i)7-s + (0.939 − 0.342i)9-s + (0.0705 + 0.122i)11-s + (−0.807 + 1.39i)13-s + (0.387 − 1.06i)15-s + 0.940·17-s + 0.501·19-s + (−0.242 − 0.289i)21-s + (−0.0110 + 0.0191i)23-s + (−0.141 − 0.244i)25-s + (−0.866 + 0.499i)27-s + (−0.815 − 1.41i)29-s + (−0.689 + 1.19i)31-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1008} (337, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1008,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$\approx$$ $$0.4969853143$$ $$L(\frac12)$$ $$\approx$$ $$0.4969853143$$ $$L(\frac{3}{2})$$ 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 + (1.70 - 0.300i)T$$
7 $$1 + (-0.5 - 0.866i)T$$
good5 $$1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2.91 - 5.04i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 - 3.87T + 17T^{2}$$
19 $$1 - 2.18T + 19T^{2}$$
23 $$1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (3.84 - 6.65i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + 7.68T + 37T^{2}$$
41 $$1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.613 - 1.06i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (2.66 + 4.61i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 0.716T + 53T^{2}$$
59 $$1 + (-0.368 + 0.637i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (4.81 - 8.34i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 13.2T + 71T^{2}$$
73 $$1 + 10.2T + 73T^{2}$$
79 $$1 + (6.31 + 10.9i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + 8.11T + 89T^{2}$$
97 $$1 + (-6.80 - 11.7i)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}