# Properties

 Label 2-490-5.4-c3-0-9 Degree $2$ Conductor $490$ Sign $0.447 - 0.894i$ Analytic cond. $28.9109$ Root an. cond. $5.37688$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2i·2-s − 7i·3-s − 4·4-s + (−10 − 5i)5-s + 14·6-s − 8i·8-s − 22·9-s + (10 − 20i)10-s − 37·11-s + 28i·12-s + 51i·13-s + (−35 + 70i)15-s + 16·16-s − 41i·17-s − 44i·18-s − 108·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 1.34i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.952·6-s − 0.353i·8-s − 0.814·9-s + (0.316 − 0.632i)10-s − 1.01·11-s + 0.673i·12-s + 1.08i·13-s + (−0.602 + 1.20i)15-s + 0.250·16-s − 0.584i·17-s − 0.576i·18-s − 1.30·19-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$490$$    =    $$2 \cdot 5 \cdot 7^{2}$$ Sign: $0.447 - 0.894i$ 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: $$(2,\ 490,\ (\ :3/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.7585708601$$ $$L(\frac12)$$ $$\approx$$ $$0.7585708601$$ $$L(\frac{5}{2})$$ 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 + (10 + 5i)T$$
7 $$1$$
good3 $$1 + 7iT - 27T^{2}$$
11 $$1 + 37T + 1.33e3T^{2}$$
13 $$1 - 51iT - 2.19e3T^{2}$$
17 $$1 + 41iT - 4.91e3T^{2}$$
19 $$1 + 108T + 6.85e3T^{2}$$
23 $$1 - 70iT - 1.21e4T^{2}$$
29 $$1 - 249T + 2.43e4T^{2}$$
31 $$1 - 134T + 2.97e4T^{2}$$
37 $$1 + 334iT - 5.06e4T^{2}$$
41 $$1 + 206T + 6.89e4T^{2}$$
43 $$1 - 376iT - 7.95e4T^{2}$$
47 $$1 - 287iT - 1.03e5T^{2}$$
53 $$1 - 6iT - 1.48e5T^{2}$$
59 $$1 + 2T + 2.05e5T^{2}$$
61 $$1 - 940T + 2.26e5T^{2}$$
67 $$1 - 106iT - 3.00e5T^{2}$$
71 $$1 - 456T + 3.57e5T^{2}$$
73 $$1 - 650iT - 3.89e5T^{2}$$
79 $$1 - 1.23e3T + 4.93e5T^{2}$$
83 $$1 - 428iT - 5.71e5T^{2}$$
89 $$1 + 220T + 7.04e5T^{2}$$
97 $$1 - 1.05e3iT - 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.88984858452741850413261644410, −9.533275396470327408956705544434, −8.390646226696373737197570646662, −7.981984847407068270269091721237, −7.03965472727530106930365939734, −6.44901021091649594988061890626, −5.12336014758603854763038433786, −4.14376528456115719513893016371, −2.45957747222641861007880517789, −0.936028658556281577130460374182, 0.30722995468467903149198132616, 2.62107990296773824875865032064, 3.50332081764238534199157002609, 4.43398132836421108305922104039, 5.19701279982007051140619635344, 6.64487809452868843139584167533, 8.242133549592249647955203628102, 8.455793591411756842923583508376, 10.06534631538059304254435294140, 10.39147450357880000793897853948