# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7$ Sign $0.667 - 0.744i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.5 + 0.866i)7-s + 5.19i·13-s + (0.5 + 0.866i)19-s + (−2.5 + 4.33i)25-s + (5.5 − 9.52i)31-s + (5.5 + 9.52i)37-s + 1.73i·43-s + (5.5 + 4.33i)49-s + (−6 + 3.46i)61-s + (13.5 + 7.79i)67-s + (1.5 + 0.866i)73-s + (4.5 − 2.59i)79-s + (−4.5 + 12.9i)91-s + 13.8i·97-s + (−6.5 − 11.2i)103-s + ⋯
 L(s)  = 1 + (0.944 + 0.327i)7-s + 1.44i·13-s + (0.114 + 0.198i)19-s + (−0.5 + 0.866i)25-s + (0.987 − 1.71i)31-s + (0.904 + 1.56i)37-s + 0.264i·43-s + (0.785 + 0.618i)49-s + (−0.768 + 0.443i)61-s + (1.64 + 0.952i)67-s + (0.175 + 0.101i)73-s + (0.506 − 0.292i)79-s + (−0.471 + 1.36i)91-s + 1.40i·97-s + (−0.640 − 1.10i)103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.667 - 0.744i$ motivic weight = $$1$$ character : $\chi_{1008} (703, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1008,\ (\ :1/2),\ 0.667 - 0.744i)$$ $$L(1)$$ $$\approx$$ $$1.713671318$$ $$L(\frac12)$$ $$\approx$$ $$1.713671318$$ $$L(\frac{3}{2})$$ 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 + (-2.5 - 0.866i)T$$
good5 $$1 + (2.5 - 4.33i)T^{2}$$
11 $$1 + (5.5 + 9.52i)T^{2}$$
13 $$1 - 5.19iT - 13T^{2}$$
17 $$1 + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 1.73iT - 43T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-13.5 - 7.79i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (44.5 - 77.0i)T^{2}$$
97 $$1 - 13.8iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}