# Properties

 Degree 2 Conductor 5 Sign $0.524 + 0.851i$ Motivic weight 8 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (15.2 − 15.2i)2-s + (−20.3 − 20.3i)3-s − 209. i·4-s + (558. + 280. i)5-s − 620.·6-s + (−2.41e3 + 2.41e3i)7-s + (705. + 705. i)8-s − 5.73e3i·9-s + (1.28e4 − 4.24e3i)10-s − 981.·11-s + (−4.26e3 + 4.26e3i)12-s + (−2.65e4 − 2.65e4i)13-s + 7.37e4i·14-s + (−5.65e3 − 1.70e4i)15-s + 7.52e4·16-s + (−1.85e4 + 1.85e4i)17-s + ⋯
 L(s)  = 1 + (0.953 − 0.953i)2-s + (−0.251 − 0.251i)3-s − 0.819i·4-s + (0.893 + 0.448i)5-s − 0.478·6-s + (−1.00 + 1.00i)7-s + (0.172 + 0.172i)8-s − 0.873i·9-s + (1.28 − 0.424i)10-s − 0.0670·11-s + (−0.205 + 0.205i)12-s + (−0.930 − 0.930i)13-s + 1.91i·14-s + (−0.111 − 0.336i)15-s + 1.14·16-s + (−0.221 + 0.221i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $0.524 + 0.851i$ motivic weight = $$8$$ character : $\chi_{5} (3, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 5,\ (\ :4),\ 0.524 + 0.851i)$$ $$L(\frac{9}{2})$$ $$\approx$$ $$1.59883 - 0.893187i$$ $$L(\frac12)$$ $$\approx$$ $$1.59883 - 0.893187i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 5$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 + (-558. - 280. i)T$$
good2 $$1 + (-15.2 + 15.2i)T - 256iT^{2}$$
3 $$1 + (20.3 + 20.3i)T + 6.56e3iT^{2}$$
7 $$1 + (2.41e3 - 2.41e3i)T - 5.76e6iT^{2}$$
11 $$1 + 981.T + 2.14e8T^{2}$$
13 $$1 + (2.65e4 + 2.65e4i)T + 8.15e8iT^{2}$$
17 $$1 + (1.85e4 - 1.85e4i)T - 6.97e9iT^{2}$$
19 $$1 - 5.03e4iT - 1.69e10T^{2}$$
23 $$1 + (-1.36e4 - 1.36e4i)T + 7.83e10iT^{2}$$
29 $$1 + 1.05e6iT - 5.00e11T^{2}$$
31 $$1 - 1.09e6T + 8.52e11T^{2}$$
37 $$1 + (-7.78e4 + 7.78e4i)T - 3.51e12iT^{2}$$
41 $$1 - 5.54e5T + 7.98e12T^{2}$$
43 $$1 + (1.07e6 + 1.07e6i)T + 1.16e13iT^{2}$$
47 $$1 + (4.06e6 - 4.06e6i)T - 2.38e13iT^{2}$$
53 $$1 + (-1.88e6 - 1.88e6i)T + 6.22e13iT^{2}$$
59 $$1 - 1.27e7iT - 1.46e14T^{2}$$
61 $$1 - 1.40e7T + 1.91e14T^{2}$$
67 $$1 + (9.54e6 - 9.54e6i)T - 4.06e14iT^{2}$$
71 $$1 + 2.82e7T + 6.45e14T^{2}$$
73 $$1 + (-1.11e7 - 1.11e7i)T + 8.06e14iT^{2}$$
79 $$1 + 6.87e7iT - 1.51e15T^{2}$$
83 $$1 + (3.29e6 + 3.29e6i)T + 2.25e15iT^{2}$$
89 $$1 - 7.97e7iT - 3.93e15T^{2}$$
97 $$1 + (1.96e7 - 1.96e7i)T - 7.83e15iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}