# Properties

 Label 2-1680-5.4-c1-0-10 Degree $2$ Conductor $1680$ Sign $0.970 - 0.241i$ Analytic cond. $13.4148$ Root an. cond. $3.66263$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s + (−2.17 + 0.539i)5-s + i·7-s − 9-s + 3.26·11-s + 0.340i·13-s + (0.539 + 2.17i)15-s − 5.75i·17-s − 6.49·19-s + 21-s + 8.49i·23-s + (4.41 − 2.34i)25-s + i·27-s + 2·29-s + 8.34·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + (−0.970 + 0.241i)5-s + 0.377i·7-s − 0.333·9-s + 0.983·11-s + 0.0943i·13-s + (0.139 + 0.560i)15-s − 1.39i·17-s − 1.49·19-s + 0.218·21-s + 1.77i·23-s + (0.883 − 0.468i)25-s + 0.192i·27-s + 0.371·29-s + 1.49·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1680$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.970 - 0.241i$ Analytic conductor: $$13.4148$$ Root analytic conductor: $$3.66263$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1680} (1009, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1680,\ (\ :1/2),\ 0.970 - 0.241i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.317539771$$ $$L(\frac12)$$ $$\approx$$ $$1.317539771$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + iT$$
5 $$1 + (2.17 - 0.539i)T$$
7 $$1 - iT$$
good11 $$1 - 3.26T + 11T^{2}$$
13 $$1 - 0.340iT - 13T^{2}$$
17 $$1 + 5.75iT - 17T^{2}$$
19 $$1 + 6.49T + 19T^{2}$$
23 $$1 - 8.49iT - 23T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 - 8.34T + 31T^{2}$$
37 $$1 - 6.15iT - 37T^{2}$$
41 $$1 - 0.340T + 41T^{2}$$
43 $$1 - 8.68iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 8.34iT - 53T^{2}$$
59 $$1 - 6.83T + 59T^{2}$$
61 $$1 - 15.3T + 61T^{2}$$
67 $$1 - 14.8iT - 67T^{2}$$
71 $$1 - 15.9T + 71T^{2}$$
73 $$1 + 1.50iT - 73T^{2}$$
79 $$1 - 8.68T + 79T^{2}$$
83 $$1 + 6.83iT - 83T^{2}$$
89 $$1 - 15.1T + 89T^{2}$$
97 $$1 - 6.49iT - 97T^{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

−9.290910101025254971465078293773, −8.412551464850948479684613823474, −7.88715050983684865817010547638, −6.80334703537802361043302113825, −6.55666637047130187775568576040, −5.26801219444348098296333370820, −4.33017784552889387201559369052, −3.38129691002454990953104965376, −2.37971119372962361395838988519, −0.955003904261396408881979855788, 0.67470077047961518219938690437, 2.32418339933843958214773565752, 3.82582147463208900487057109343, 4.06470651024197752270995687351, 4.94900608211865650205103718124, 6.30494756283707991613294308451, 6.73957531082492283286122006080, 8.049120421527629819215054145356, 8.476620154307477120045604674825, 9.150215148448318456492251630144