# Properties

 Degree 2 Conductor 5 Sign $0.866 + 0.499i$ Motivic weight 17 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 275. i·2-s − 2.99e3i·3-s + 5.52e4·4-s + (−7.56e5 − 4.36e5i)5-s + 8.23e5·6-s − 2.43e7i·7-s + 5.13e7i·8-s + 1.20e8·9-s + (1.20e8 − 2.08e8i)10-s + 8.41e8·11-s − 1.65e8i·12-s − 4.01e9i·13-s + 6.71e9·14-s + (−1.30e9 + 2.26e9i)15-s − 6.88e9·16-s + 8.60e9i·17-s + ⋯
 L(s)  = 1 + 0.760i·2-s − 0.263i·3-s + 0.421·4-s + (−0.866 − 0.499i)5-s + 0.200·6-s − 1.59i·7-s + 1.08i·8-s + 0.930·9-s + (0.379 − 0.658i)10-s + 1.18·11-s − 0.110i·12-s − 1.36i·13-s + 1.21·14-s + (−0.131 + 0.227i)15-s − 0.400·16-s + 0.299i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(18-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $0.866 + 0.499i$ motivic weight = $$17$$ character : $\chi_{5} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 5,\ (\ :17/2),\ 0.866 + 0.499i)$ $L(9)$ $\approx$ $1.71357 - 0.458674i$ $L(\frac12)$ $\approx$ $1.71357 - 0.458674i$ $L(\frac{19}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ 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 + (7.56e5 + 4.36e5i)T$$
good2 $$1 - 275. iT - 1.31e5T^{2}$$
3 $$1 + 2.99e3iT - 1.29e8T^{2}$$
7 $$1 + 2.43e7iT - 2.32e14T^{2}$$
11 $$1 - 8.41e8T + 5.05e17T^{2}$$
13 $$1 + 4.01e9iT - 8.65e18T^{2}$$
17 $$1 - 8.60e9iT - 8.27e20T^{2}$$
19 $$1 + 6.64e10T + 5.48e21T^{2}$$
23 $$1 + 5.12e11iT - 1.41e23T^{2}$$
29 $$1 - 1.63e11T + 7.25e24T^{2}$$
31 $$1 + 3.12e12T + 2.25e25T^{2}$$
37 $$1 + 2.41e12iT - 4.56e26T^{2}$$
41 $$1 - 4.43e13T + 2.61e27T^{2}$$
43 $$1 - 7.24e13iT - 5.87e27T^{2}$$
47 $$1 - 5.71e13iT - 2.66e28T^{2}$$
53 $$1 - 3.35e14iT - 2.05e29T^{2}$$
59 $$1 + 2.43e14T + 1.27e30T^{2}$$
61 $$1 - 2.41e15T + 2.24e30T^{2}$$
67 $$1 - 2.63e15iT - 1.10e31T^{2}$$
71 $$1 - 4.51e15T + 2.96e31T^{2}$$
73 $$1 - 2.61e15iT - 4.74e31T^{2}$$
79 $$1 - 7.48e15T + 1.81e32T^{2}$$
83 $$1 - 2.07e15iT - 4.21e32T^{2}$$
89 $$1 - 2.77e15T + 1.37e33T^{2}$$
97 $$1 - 5.77e16iT - 5.95e33T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}