# Properties

 Degree 2 Conductor $2^{3} \cdot 3^{2} \cdot 7^{2}$ Sign $-0.991 + 0.126i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − 1.73i)5-s + (−1 + 1.73i)11-s + 2·13-s + (−3 + 5.19i)17-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + (2 − 3.46i)31-s + (−5 − 8.66i)37-s + 2·41-s − 4·43-s + (−2 − 3.46i)47-s + (6 − 10.3i)53-s + 3.99·55-s + (−6 + 10.3i)59-s + ⋯
 L(s)  = 1 + (−0.447 − 0.774i)5-s + (−0.301 + 0.522i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + (0.359 − 0.622i)31-s + (−0.821 − 1.42i)37-s + 0.312·41-s − 0.609·43-s + (−0.291 − 0.505i)47-s + (0.824 − 1.42i)53-s + 0.539·55-s + (−0.781 + 1.35i)59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3528$$    =    $$2^{3} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $-0.991 + 0.126i$ motivic weight = $$1$$ character : $\chi_{3528} (361, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3528,\ (\ :1/2),\ -0.991 + 0.126i)$$ $$L(1)$$ $$\approx$$ $$0.3209722599$$ $$L(\frac12)$$ $$\approx$$ $$0.3209722599$$ $$L(\frac{3}{2})$$ 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$$
good5 $$1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 14T + 71T^{2}$$
73 $$1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 16T + 83T^{2}$$
89 $$1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 18T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}