# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7$ Sign $-0.0633 + 0.997i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.35 + 1.93i)5-s + (−2.5 + 0.866i)7-s + (3.35 + 1.93i)11-s − 3.46i·13-s + (−2 − 3.46i)19-s + (−6.70 + 3.87i)23-s + (5.00 − 8.66i)25-s + 6.70·29-s + (0.5 − 0.866i)31-s + (6.70 − 7.74i)35-s + (−2 − 3.46i)37-s − 7.74i·41-s − 6.92i·43-s + (−6.70 − 11.6i)47-s + (5.5 − 4.33i)49-s + ⋯
 L(s)  = 1 + (−1.50 + 0.866i)5-s + (−0.944 + 0.327i)7-s + (1.01 + 0.583i)11-s − 0.960i·13-s + (−0.458 − 0.794i)19-s + (−1.39 + 0.807i)23-s + (1.00 − 1.73i)25-s + 1.24·29-s + (0.0898 − 0.155i)31-s + (1.13 − 1.30i)35-s + (−0.328 − 0.569i)37-s − 1.20i·41-s − 1.05i·43-s + (−0.978 − 1.69i)47-s + (0.785 − 0.618i)49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.0633 + 0.997i$ motivic weight = $$1$$ character : $\chi_{1008} (703, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1008,\ (\ :1/2),\ -0.0633 + 0.997i)$$ $$L(1)$$ $$\approx$$ $$0.4555432341$$ $$L(\frac12)$$ $$\approx$$ $$0.4555432341$$ $$L(\frac{3}{2})$$ 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 + (3.35 - 1.93i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-3.35 - 1.93i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 + (8.5 + 14.7i)T^{2}$$
19 $$1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (6.70 - 3.87i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 - 6.70T + 29T^{2}$$
31 $$1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 7.74iT - 41T^{2}$$
43 $$1 + 6.92iT - 43T^{2}$$
47 $$1 + (6.70 + 11.6i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-3.35 + 5.80i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-6 - 3.46i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 - 7.74iT - 71T^{2}$$
73 $$1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 - 6.70T + 83T^{2}$$
89 $$1 + (6.70 - 3.87i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 - 5.19iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}