# Properties

 Label 2-490-5.4-c3-0-55 Degree $2$ Conductor $490$ Sign $-0.894 - 0.447i$ Analytic cond. $28.9109$ Root an. cond. $5.37688$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·2-s − 2i·3-s − 4·4-s + (5 − 10i)5-s − 4·6-s + 8i·8-s + 23·9-s + (−20 − 10i)10-s − 28·11-s + 8i·12-s − 12i·13-s + (−20 − 10i)15-s + 16·16-s − 64i·17-s − 46i·18-s − 60·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.384i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s − 0.272·6-s + 0.353i·8-s + 0.851·9-s + (−0.632 − 0.316i)10-s − 0.767·11-s + 0.192i·12-s − 0.256i·13-s + (−0.344 − 0.172i)15-s + 0.250·16-s − 0.913i·17-s − 0.602i·18-s − 0.724·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$490$$    =    $$2 \cdot 5 \cdot 7^{2}$$ Sign: $-0.894 - 0.447i$ Analytic conductor: $$28.9109$$ Root analytic conductor: $$5.37688$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{490} (99, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 490,\ (\ :3/2),\ -0.894 - 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.273234713$$ $$L(\frac12)$$ $$\approx$$ $$1.273234713$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2iT$$
5 $$1 + (-5 + 10i)T$$
7 $$1$$
good3 $$1 + 2iT - 27T^{2}$$
11 $$1 + 28T + 1.33e3T^{2}$$
13 $$1 + 12iT - 2.19e3T^{2}$$
17 $$1 + 64iT - 4.91e3T^{2}$$
19 $$1 + 60T + 6.85e3T^{2}$$
23 $$1 + 58iT - 1.21e4T^{2}$$
29 $$1 + 90T + 2.43e4T^{2}$$
31 $$1 - 128T + 2.97e4T^{2}$$
37 $$1 + 236iT - 5.06e4T^{2}$$
41 $$1 + 242T + 6.89e4T^{2}$$
43 $$1 - 362iT - 7.95e4T^{2}$$
47 $$1 - 226iT - 1.03e5T^{2}$$
53 $$1 + 108iT - 1.48e5T^{2}$$
59 $$1 + 20T + 2.05e5T^{2}$$
61 $$1 + 542T + 2.26e5T^{2}$$
67 $$1 - 434iT - 3.00e5T^{2}$$
71 $$1 + 1.12e3T + 3.57e5T^{2}$$
73 $$1 + 632iT - 3.89e5T^{2}$$
79 $$1 - 720T + 4.93e5T^{2}$$
83 $$1 - 478iT - 5.71e5T^{2}$$
89 $$1 + 490T + 7.04e5T^{2}$$
97 $$1 - 1.45e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$