# Properties

 Degree 2 Conductor $2^{3} \cdot 3^{3} \cdot 7$ Sign $0.187 + 0.982i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.119 + 0.207i)5-s + (0.710 − 2.54i)7-s + (1.21 − 2.09i)11-s + 0.760·13-s + (−1.71 + 2.96i)17-s + (−0.590 − 1.02i)19-s + (−1.09 − 1.88i)23-s + (2.47 − 4.28i)25-s + 2.89·29-s + (2.32 − 4.02i)31-s + (0.612 − 0.157i)35-s + (−2.89 − 5.00i)37-s − 8.54·41-s − 3.37·43-s + (2.58 + 4.47i)47-s + ⋯
 L(s)  = 1 + (0.0534 + 0.0926i)5-s + (0.268 − 0.963i)7-s + (0.364 − 0.632i)11-s + 0.211·13-s + (−0.414 + 0.718i)17-s + (−0.135 − 0.234i)19-s + (−0.227 − 0.394i)23-s + (0.494 − 0.856i)25-s + 0.538·29-s + (0.417 − 0.722i)31-s + (0.103 − 0.0266i)35-s + (−0.475 − 0.823i)37-s − 1.33·41-s − 0.515·43-s + (0.376 + 0.652i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1512$$    =    $$2^{3} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.187 + 0.982i$ motivic weight = $$1$$ character : $\chi_{1512} (865, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1512,\ (\ :1/2),\ 0.187 + 0.982i)$$ $$L(1)$$ $$\approx$$ $$1.571964027$$ $$L(\frac12)$$ $$\approx$$ $$1.571964027$$ $$L(\frac{3}{2})$$ 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 + (-0.710 + 2.54i)T$$
good5 $$1 + (-0.119 - 0.207i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-1.21 + 2.09i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 0.760T + 13T^{2}$$
17 $$1 + (1.71 - 2.96i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (0.590 + 1.02i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (1.09 + 1.88i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 2.89T + 29T^{2}$$
31 $$1 + (-2.32 + 4.02i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (2.89 + 5.00i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 8.54T + 41T^{2}$$
43 $$1 + 3.37T + 43T^{2}$$
47 $$1 + (-2.58 - 4.47i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (1.56 - 2.70i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-3.11 + 5.39i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (0.681 + 1.18i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-7.03 + 12.1i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 8.02T + 71T^{2}$$
73 $$1 + (-3.91 + 6.77i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (2.27 + 3.93i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 12.0T + 83T^{2}$$
89 $$1 + (2.73 + 4.74i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 10.3T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}