# 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 + 1.82i·2-s + 3i·3-s + 4.65·4-s − 5.48·6-s − 7i·7-s + 23.1i·8-s − 9·9-s − 64.5·11-s + 13.9i·12-s + 32.3i·13-s + 12.7·14-s − 5.05·16-s − 56.3i·17-s − 16.4i·18-s + 2.74·19-s + ⋯
 L(s)  = 1 + 0.646i·2-s + 0.577i·3-s + 0.582·4-s − 0.373·6-s − 0.377i·7-s + 1.02i·8-s − 0.333·9-s − 1.76·11-s + 0.336i·12-s + 0.690i·13-s + 0.244·14-s − 0.0790·16-s − 0.803i·17-s − 0.215i·18-s + 0.0331·19-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 = $$(2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)$$ $$L(2)$$ $$\approx$$ $$0.2904210379$$ $$L(\frac12)$$ $$\approx$$ $$0.2904210379$$ $$L(\frac{5}{2})$$ 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 - 1.82iT - 8T^{2}$$
11 $$1 + 64.5T + 1.33e3T^{2}$$
13 $$1 - 32.3iT - 2.19e3T^{2}$$
17 $$1 + 56.3iT - 4.91e3T^{2}$$
19 $$1 - 2.74T + 6.85e3T^{2}$$
23 $$1 + 88.1iT - 1.21e4T^{2}$$
29 $$1 + 246.T + 2.43e4T^{2}$$
31 $$1 + 110.T + 2.97e4T^{2}$$
37 $$1 - 120. iT - 5.06e4T^{2}$$
41 $$1 + 176.T + 6.89e4T^{2}$$
43 $$1 - 443. iT - 7.95e4T^{2}$$
47 $$1 + 345. iT - 1.03e5T^{2}$$
53 $$1 + 260. iT - 1.48e5T^{2}$$
59 $$1 + 628.T + 2.05e5T^{2}$$
61 $$1 + 115.T + 2.26e5T^{2}$$
67 $$1 + 951. iT - 3.00e5T^{2}$$
71 $$1 - 356.T + 3.57e5T^{2}$$
73 $$1 - 656. iT - 3.89e5T^{2}$$
79 $$1 + 440.T + 4.93e5T^{2}$$
83 $$1 - 54.4iT - 5.71e5T^{2}$$
89 $$1 - 1.01e3T + 7.04e5T^{2}$$
97 $$1 + 724. iT - 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}