# Properties

 Degree 2 Conductor 5 Sign $-0.814 + 0.580i$ Motivic weight 9 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 41.3i·2-s + 37.6i·3-s − 1.19e3·4-s + (1.13e3 − 810. i)5-s + 1.55e3·6-s − 5.31e3i·7-s + 2.82e4i·8-s + 1.82e4·9-s + (−3.35e4 − 4.70e4i)10-s + 1.04e4·11-s − 4.49e4i·12-s + 7.96e4i·13-s − 2.19e5·14-s + (3.05e4 + 4.28e4i)15-s + 5.54e5·16-s + 3.13e5i·17-s + ⋯
 L(s)  = 1 − 1.82i·2-s + 0.268i·3-s − 2.33·4-s + (0.814 − 0.580i)5-s + 0.489·6-s − 0.836i·7-s + 2.43i·8-s + 0.928·9-s + (−1.05 − 1.48i)10-s + 0.214·11-s − 0.626i·12-s + 0.773i·13-s − 1.52·14-s + (0.155 + 0.218i)15-s + 2.11·16-s + 0.911i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $-0.814 + 0.580i$ motivic weight = $$9$$ character : $\chi_{5} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 5,\ (\ :9/2),\ -0.814 + 0.580i)$ $L(5)$ $\approx$ $0.401675 - 1.25623i$ $L(\frac12)$ $\approx$ $0.401675 - 1.25623i$ $L(\frac{11}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1 + (-1.13e3 + 810. i)T$$
good2 $$1 + 41.3iT - 512T^{2}$$
3 $$1 - 37.6iT - 1.96e4T^{2}$$
7 $$1 + 5.31e3iT - 4.03e7T^{2}$$
11 $$1 - 1.04e4T + 2.35e9T^{2}$$
13 $$1 - 7.96e4iT - 1.06e10T^{2}$$
17 $$1 - 3.13e5iT - 1.18e11T^{2}$$
19 $$1 - 2.46e5T + 3.22e11T^{2}$$
23 $$1 + 7.21e5iT - 1.80e12T^{2}$$
29 $$1 + 2.56e6T + 1.45e13T^{2}$$
31 $$1 + 3.29e6T + 2.64e13T^{2}$$
37 $$1 - 1.40e7iT - 1.29e14T^{2}$$
41 $$1 - 1.70e7T + 3.27e14T^{2}$$
43 $$1 + 2.92e7iT - 5.02e14T^{2}$$
47 $$1 - 4.10e7iT - 1.11e15T^{2}$$
53 $$1 - 5.67e7iT - 3.29e15T^{2}$$
59 $$1 + 1.60e8T + 8.66e15T^{2}$$
61 $$1 - 5.33e7T + 1.16e16T^{2}$$
67 $$1 + 2.80e8iT - 2.72e16T^{2}$$
71 $$1 + 8.97e7T + 4.58e16T^{2}$$
73 $$1 - 7.60e7iT - 5.88e16T^{2}$$
79 $$1 + 4.10e8T + 1.19e17T^{2}$$
83 $$1 - 5.21e8iT - 1.86e17T^{2}$$
89 $$1 + 2.37e8T + 3.50e17T^{2}$$
97 $$1 + 6.03e8iT - 7.60e17T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}