# Properties

 Degree $2$ Conductor $5$ Sign $0.0306 - 0.999i$ Motivic weight $32$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.20e4 + 4.20e4i)2-s + (−2.23e7 − 2.23e7i)3-s + 7.62e8i·4-s + (−7.62e10 − 1.32e11i)5-s + 1.87e12·6-s + (−2.39e13 + 2.39e13i)7-s + (−2.12e14 − 2.12e14i)8-s − 8.55e14i·9-s + (8.75e15 + 2.35e15i)10-s − 3.17e16·11-s + (1.70e16 − 1.70e16i)12-s + (−5.32e17 − 5.32e17i)13-s − 2.01e18i·14-s + (−1.25e18 + 4.65e18i)15-s + 1.45e19·16-s + (4.10e19 − 4.10e19i)17-s + ⋯
 L(s)  = 1 + (−0.641 + 0.641i)2-s + (−0.518 − 0.518i)3-s + 0.177i·4-s + (−0.499 − 0.866i)5-s + 0.665·6-s + (−0.721 + 0.721i)7-s + (−0.755 − 0.755i)8-s − 0.461i·9-s + (0.875 + 0.235i)10-s − 0.690·11-s + (0.0920 − 0.0920i)12-s + (−0.800 − 0.800i)13-s − 0.925i·14-s + (−0.190 + 0.708i)15-s + 0.791·16-s + (0.842 − 0.842i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5$$ Sign: $0.0306 - 0.999i$ Motivic weight: $$32$$ Character: $\chi_{5} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5,\ (\ :16),\ 0.0306 - 0.999i)$$

## Particular Values

 $$L(\frac{33}{2})$$ $$\approx$$ $$0.1823529760$$ $$L(\frac12)$$ $$\approx$$ $$0.1823529760$$ $$L(17)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (7.62e10 + 1.32e11i)T$$
good2 $$1 + (4.20e4 - 4.20e4i)T - 4.29e9iT^{2}$$
3 $$1 + (2.23e7 + 2.23e7i)T + 1.85e15iT^{2}$$
7 $$1 + (2.39e13 - 2.39e13i)T - 1.10e27iT^{2}$$
11 $$1 + 3.17e16T + 2.11e33T^{2}$$
13 $$1 + (5.32e17 + 5.32e17i)T + 4.42e35iT^{2}$$
17 $$1 + (-4.10e19 + 4.10e19i)T - 2.36e39iT^{2}$$
19 $$1 + 2.22e20iT - 8.31e40T^{2}$$
23 $$1 + (6.22e21 + 6.22e21i)T + 3.76e43iT^{2}$$
29 $$1 - 3.25e23iT - 6.26e46T^{2}$$
31 $$1 - 1.21e22T + 5.29e47T^{2}$$
37 $$1 + (1.53e25 - 1.53e25i)T - 1.52e50iT^{2}$$
41 $$1 + 2.70e25T + 4.06e51T^{2}$$
43 $$1 + (-9.54e25 - 9.54e25i)T + 1.86e52iT^{2}$$
47 $$1 + (-2.53e26 + 2.53e26i)T - 3.21e53iT^{2}$$
53 $$1 + (1.49e27 + 1.49e27i)T + 1.50e55iT^{2}$$
59 $$1 + 2.48e28iT - 4.64e56T^{2}$$
61 $$1 - 3.17e28T + 1.35e57T^{2}$$
67 $$1 + (3.16e27 - 3.16e27i)T - 2.71e58iT^{2}$$
71 $$1 - 4.69e29T + 1.73e59T^{2}$$
73 $$1 + (3.04e29 + 3.04e29i)T + 4.22e59iT^{2}$$
79 $$1 - 1.79e30iT - 5.29e60T^{2}$$
83 $$1 + (4.68e29 + 4.68e29i)T + 2.57e61iT^{2}$$
89 $$1 - 2.97e31iT - 2.40e62T^{2}$$
97 $$1 + (1.89e31 - 1.89e31i)T - 3.77e63iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$