# Properties

 Degree 2 Conductor 5 Sign $0.0242 + 0.999i$ Motivic weight 17 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 660. i·2-s + 1.13e4i·3-s − 3.05e5·4-s + (−2.11e4 − 8.73e5i)5-s − 7.47e6·6-s + 2.37e7i·7-s − 1.15e8i·8-s + 1.24e6·9-s + (5.76e8 − 1.39e7i)10-s − 3.40e8·11-s − 3.45e9i·12-s − 6.20e8i·13-s − 1.56e10·14-s + (9.87e9 − 2.39e8i)15-s + 3.59e10·16-s + 9.78e9i·17-s + ⋯
 L(s)  = 1 + 1.82i·2-s + 0.995i·3-s − 2.32·4-s + (−0.0242 − 0.999i)5-s − 1.81·6-s + 1.55i·7-s − 2.42i·8-s + 0.00960·9-s + (1.82 − 0.0442i)10-s − 0.478·11-s − 2.31i·12-s − 0.210i·13-s − 2.83·14-s + (0.994 − 0.0241i)15-s + 2.09·16-s + 0.340i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(18-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $0.0242 + 0.999i$ motivic weight = $$17$$ character : $\chi_{5} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 5,\ (\ :17/2),\ 0.0242 + 0.999i)$ $L(9)$ $\approx$ $0.661618 - 0.645759i$ $L(\frac12)$ $\approx$ $0.661618 - 0.645759i$ $L(\frac{19}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p(T)$$ 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 + (2.11e4 + 8.73e5i)T$$
good2 $$1 - 660. iT - 1.31e5T^{2}$$
3 $$1 - 1.13e4iT - 1.29e8T^{2}$$
7 $$1 - 2.37e7iT - 2.32e14T^{2}$$
11 $$1 + 3.40e8T + 5.05e17T^{2}$$
13 $$1 + 6.20e8iT - 8.65e18T^{2}$$
17 $$1 - 9.78e9iT - 8.27e20T^{2}$$
19 $$1 + 3.92e10T + 5.48e21T^{2}$$
23 $$1 + 2.44e11iT - 1.41e23T^{2}$$
29 $$1 + 3.98e12T + 7.25e24T^{2}$$
31 $$1 - 2.84e12T + 2.25e25T^{2}$$
37 $$1 - 2.87e13iT - 4.56e26T^{2}$$
41 $$1 - 2.58e13T + 2.61e27T^{2}$$
43 $$1 - 7.92e13iT - 5.87e27T^{2}$$
47 $$1 - 3.42e13iT - 2.66e28T^{2}$$
53 $$1 - 6.28e14iT - 2.05e29T^{2}$$
59 $$1 - 1.43e15T + 1.27e30T^{2}$$
61 $$1 + 1.41e15T + 2.24e30T^{2}$$
67 $$1 + 3.52e14iT - 1.10e31T^{2}$$
71 $$1 - 5.76e15T + 2.96e31T^{2}$$
73 $$1 + 1.24e15iT - 4.74e31T^{2}$$
79 $$1 + 8.16e15T + 1.81e32T^{2}$$
83 $$1 - 2.09e16iT - 4.21e32T^{2}$$
89 $$1 + 2.91e16T + 1.37e33T^{2}$$
97 $$1 + 4.58e16iT - 5.95e33T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}