# Properties

 Degree 2 Conductor $2^{3} \cdot 3^{2} \cdot 7$ Sign $0.947 + 0.320i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.835 + 1.14i)2-s + (−0.602 − 1.90i)4-s + 0.467i·5-s + 7-s + (2.67 + 0.907i)8-s + (−0.532 − 0.390i)10-s − 4.87i·11-s − 4.56i·13-s + (−0.835 + 1.14i)14-s + (−3.27 + 2.29i)16-s − 6.09·17-s + 1.34i·19-s + (0.890 − 0.281i)20-s + (5.56 + 4.07i)22-s + 4.09·23-s + ⋯
 L(s)  = 1 + (−0.591 + 0.806i)2-s + (−0.301 − 0.953i)4-s + 0.208i·5-s + 0.377·7-s + (0.947 + 0.320i)8-s + (−0.168 − 0.123i)10-s − 1.47i·11-s − 1.26i·13-s + (−0.223 + 0.304i)14-s + (−0.818 + 0.574i)16-s − 1.47·17-s + 0.308i·19-s + (0.199 − 0.0629i)20-s + (1.18 + 0.869i)22-s + 0.854·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$504$$    =    $$2^{3} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.947 + 0.320i$ motivic weight = $$1$$ character : $\chi_{504} (253, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 504,\ (\ :1/2),\ 0.947 + 0.320i)$$ $$L(1)$$ $$\approx$$ $$0.946389 - 0.155879i$$ $$L(\frac12)$$ $$\approx$$ $$0.946389 - 0.155879i$$ $$L(\frac{3}{2})$$ 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 + (0.835 - 1.14i)T$$
3 $$1$$
7 $$1 - T$$
good5 $$1 - 0.467iT - 5T^{2}$$
11 $$1 + 4.87iT - 11T^{2}$$
13 $$1 + 4.56iT - 13T^{2}$$
17 $$1 + 6.09T + 17T^{2}$$
19 $$1 - 1.34iT - 19T^{2}$$
23 $$1 - 4.09T + 23T^{2}$$
29 $$1 + 7.78iT - 29T^{2}$$
31 $$1 - 4.40T + 31T^{2}$$
37 $$1 + 4.40iT - 37T^{2}$$
41 $$1 - 6.09T + 41T^{2}$$
43 $$1 - 4.15iT - 43T^{2}$$
47 $$1 - 6.68T + 47T^{2}$$
53 $$1 - 1.34iT - 53T^{2}$$
59 $$1 + 4iT - 59T^{2}$$
61 $$1 + 5.49iT - 61T^{2}$$
67 $$1 - 5.90iT - 67T^{2}$$
71 $$1 - 4.72T + 71T^{2}$$
73 $$1 + 12.0T + 73T^{2}$$
79 $$1 + 16.1T + 79T^{2}$$
83 $$1 - 13.7iT - 83T^{2}$$
89 $$1 + 7.96T + 89T^{2}$$
97 $$1 + 12.8T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}