Properties

 Degree $2$ Conductor $5$ Sign $-0.969 - 0.246i$ Motivic weight $8$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−11.8 − 11.8i)2-s + (−90.9 + 90.9i)3-s + 25.5i·4-s + (−434. − 449. i)5-s + 2.15e3·6-s + (508. + 508. i)7-s + (−2.73e3 + 2.73e3i)8-s − 9.98e3i·9-s + (−178. + 1.04e4i)10-s − 7.02e3·11-s + (−2.32e3 − 2.32e3i)12-s + (−8.07e3 + 8.07e3i)13-s − 1.20e4i·14-s + (8.03e4 + 1.36e3i)15-s + 7.14e4·16-s + (−1.02e5 − 1.02e5i)17-s + ⋯
 L(s)  = 1 + (−0.741 − 0.741i)2-s + (−1.12 + 1.12i)3-s + 0.0998i·4-s + (−0.694 − 0.719i)5-s + 1.66·6-s + (0.211 + 0.211i)7-s + (−0.667 + 0.667i)8-s − 1.52i·9-s + (−0.0178 + 1.04i)10-s − 0.479·11-s + (−0.112 − 0.112i)12-s + (−0.282 + 0.282i)13-s − 0.313i·14-s + (1.58 + 0.0269i)15-s + 1.08·16-s + (−1.23 − 1.23i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$5$$ Sign: $-0.969 - 0.246i$ Motivic weight: $$8$$ Character: $\chi_{5} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5,\ (\ :4),\ -0.969 - 0.246i)$$

Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.00665038 + 0.0531824i$$ $$L(\frac12)$$ $$\approx$$ $$0.00665038 + 0.0531824i$$ $$L(5)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (434. + 449. i)T$$
good2 $$1 + (11.8 + 11.8i)T + 256iT^{2}$$
3 $$1 + (90.9 - 90.9i)T - 6.56e3iT^{2}$$
7 $$1 + (-508. - 508. i)T + 5.76e6iT^{2}$$
11 $$1 + 7.02e3T + 2.14e8T^{2}$$
13 $$1 + (8.07e3 - 8.07e3i)T - 8.15e8iT^{2}$$
17 $$1 + (1.02e5 + 1.02e5i)T + 6.97e9iT^{2}$$
19 $$1 - 5.95e4iT - 1.69e10T^{2}$$
23 $$1 + (-1.32e5 + 1.32e5i)T - 7.83e10iT^{2}$$
29 $$1 - 3.92e5iT - 5.00e11T^{2}$$
31 $$1 + 5.07e5T + 8.52e11T^{2}$$
37 $$1 + (6.10e4 + 6.10e4i)T + 3.51e12iT^{2}$$
41 $$1 + 1.81e6T + 7.98e12T^{2}$$
43 $$1 + (-1.47e6 + 1.47e6i)T - 1.16e13iT^{2}$$
47 $$1 + (1.79e6 + 1.79e6i)T + 2.38e13iT^{2}$$
53 $$1 + (5.66e6 - 5.66e6i)T - 6.22e13iT^{2}$$
59 $$1 - 1.74e7iT - 1.46e14T^{2}$$
61 $$1 + 1.96e7T + 1.91e14T^{2}$$
67 $$1 + (1.12e7 + 1.12e7i)T + 4.06e14iT^{2}$$
71 $$1 - 3.01e7T + 6.45e14T^{2}$$
73 $$1 + (-2.52e7 + 2.52e7i)T - 8.06e14iT^{2}$$
79 $$1 - 8.14e6iT - 1.51e15T^{2}$$
83 $$1 + (1.99e7 - 1.99e7i)T - 2.25e15iT^{2}$$
89 $$1 + 8.20e7iT - 3.93e15T^{2}$$
97 $$1 + (-3.37e7 - 3.37e7i)T + 7.83e15iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$