# Properties

 Degree 2 Conductor $3 \cdot 5^{2} \cdot 7$ Sign $-0.447 - 0.894i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.82i·2-s − 3i·3-s − 6.65·4-s + 11.4·6-s + 7i·7-s + 5.14i·8-s − 9·9-s + 48.5·11-s + 19.9i·12-s − 43.6i·13-s − 26.7·14-s − 72.9·16-s + 67.6i·17-s − 34.4i·18-s + 93.2·19-s + ⋯
 L(s)  = 1 + 1.35i·2-s − 0.577i·3-s − 0.832·4-s + 0.781·6-s + 0.377i·7-s + 0.227i·8-s − 0.333·9-s + 1.33·11-s + 0.480i·12-s − 0.931i·13-s − 0.511·14-s − 1.13·16-s + 0.965i·17-s − 0.451i·18-s + 1.12·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$525$$    =    $$3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.447 - 0.894i$ motivic weight = $$3$$ character : $\chi_{525} (274, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 525,\ (\ :3/2),\ -0.447 - 0.894i)$$ $$L(2)$$ $$\approx$$ $$2.065439991$$ $$L(\frac12)$$ $$\approx$$ $$2.065439991$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + 3iT$$
5 $$1$$
7 $$1 - 7iT$$
good2 $$1 - 3.82iT - 8T^{2}$$
11 $$1 - 48.5T + 1.33e3T^{2}$$
13 $$1 + 43.6iT - 2.19e3T^{2}$$
17 $$1 - 67.6iT - 4.91e3T^{2}$$
19 $$1 - 93.2T + 6.85e3T^{2}$$
23 $$1 + 104. iT - 1.21e4T^{2}$$
29 $$1 - 58.7T + 2.43e4T^{2}$$
31 $$1 + 9.08T + 2.97e4T^{2}$$
37 $$1 - 252. iT - 5.06e4T^{2}$$
41 $$1 - 276.T + 6.89e4T^{2}$$
43 $$1 + 92.6iT - 7.95e4T^{2}$$
47 $$1 - 582. iT - 1.03e5T^{2}$$
53 $$1 - 623. iT - 1.48e5T^{2}$$
59 $$1 - 524.T + 2.05e5T^{2}$$
61 $$1 + 352.T + 2.26e5T^{2}$$
67 $$1 - 736. iT - 3.00e5T^{2}$$
71 $$1 + 492.T + 3.57e5T^{2}$$
73 $$1 - 1.16e3iT - 3.89e5T^{2}$$
79 $$1 - 872.T + 4.93e5T^{2}$$
83 $$1 + 529. iT - 5.71e5T^{2}$$
89 $$1 - 385.T + 7.04e5T^{2}$$
97 $$1 - 463. iT - 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}