# Properties

 Degree $2$ Conductor $10$ Sign $0.178 - 0.983i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s + 14i·3-s − 16·4-s + (55 + 10i)5-s − 56·6-s − 158i·7-s − 64i·8-s + 47·9-s + (−40 + 220i)10-s − 148·11-s − 224i·12-s + 684i·13-s + 632·14-s + (−140 + 770i)15-s + 256·16-s − 2.04e3i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.898i·3-s − 0.5·4-s + (0.983 + 0.178i)5-s − 0.635·6-s − 1.21i·7-s − 0.353i·8-s + 0.193·9-s + (−0.126 + 0.695i)10-s − 0.368·11-s − 0.449i·12-s + 1.12i·13-s + 0.861·14-s + (−0.160 + 0.883i)15-s + 0.250·16-s − 1.71i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$10$$    =    $$2 \cdot 5$$ Sign: $0.178 - 0.983i$ Motivic weight: $$5$$ Character: $\chi_{10} (9, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 10,\ (\ :5/2),\ 0.178 - 0.983i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.950751 + 0.793474i$$ $$L(\frac12)$$ $$\approx$$ $$0.950751 + 0.793474i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 4iT$$
5 $$1 + (-55 - 10i)T$$
good3 $$1 - 14iT - 243T^{2}$$
7 $$1 + 158iT - 1.68e4T^{2}$$
11 $$1 + 148T + 1.61e5T^{2}$$
13 $$1 - 684iT - 3.71e5T^{2}$$
17 $$1 + 2.04e3iT - 1.41e6T^{2}$$
19 $$1 + 2.22e3T + 2.47e6T^{2}$$
23 $$1 + 1.24e3iT - 6.43e6T^{2}$$
29 $$1 - 270T + 2.05e7T^{2}$$
31 $$1 + 2.04e3T + 2.86e7T^{2}$$
37 $$1 - 4.37e3iT - 6.93e7T^{2}$$
41 $$1 + 2.39e3T + 1.15e8T^{2}$$
43 $$1 - 2.29e3iT - 1.47e8T^{2}$$
47 $$1 - 1.06e4iT - 2.29e8T^{2}$$
53 $$1 - 2.96e3iT - 4.18e8T^{2}$$
59 $$1 - 3.97e4T + 7.14e8T^{2}$$
61 $$1 + 4.22e4T + 8.44e8T^{2}$$
67 $$1 + 3.20e4iT - 1.35e9T^{2}$$
71 $$1 + 4.24e3T + 1.80e9T^{2}$$
73 $$1 - 3.01e4iT - 2.07e9T^{2}$$
79 $$1 + 3.52e4T + 3.07e9T^{2}$$
83 $$1 + 2.78e4iT - 3.93e9T^{2}$$
89 $$1 - 8.52e4T + 5.58e9T^{2}$$
97 $$1 - 9.72e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$