# Properties

 Degree $2$ Conductor $1680$ Sign $-0.975 + 0.218i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.618 + 1.61i)3-s − 5-s + (−0.381 + 2.61i)7-s + (−2.23 + 2.00i)9-s + 4.47i·11-s − 3.23i·13-s + (−0.618 − 1.61i)15-s − 0.763·17-s − 0.472i·19-s + (−4.47 + 1.00i)21-s + 4i·23-s + 25-s + (−4.61 − 2.38i)27-s + 5.70i·29-s − 7.23i·31-s + ⋯
 L(s)  = 1 + (0.356 + 0.934i)3-s − 0.447·5-s + (−0.144 + 0.989i)7-s + (−0.745 + 0.666i)9-s + 1.34i·11-s − 0.897i·13-s + (−0.159 − 0.417i)15-s − 0.185·17-s − 0.108i·19-s + (−0.975 + 0.218i)21-s + 0.834i·23-s + 0.200·25-s + (−0.888 − 0.458i)27-s + 1.05i·29-s − 1.29i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1680$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.975 + 0.218i$ Motivic weight: $$1$$ Character: $\chi_{1680} (881, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1680,\ (\ :1/2),\ -0.975 + 0.218i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9374554259$$ $$L(\frac12)$$ $$\approx$$ $$0.9374554259$$ $$L(\frac{3}{2})$$ 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 + (-0.618 - 1.61i)T$$
5 $$1 + T$$
7 $$1 + (0.381 - 2.61i)T$$
good11 $$1 - 4.47iT - 11T^{2}$$
13 $$1 + 3.23iT - 13T^{2}$$
17 $$1 + 0.763T + 17T^{2}$$
19 $$1 + 0.472iT - 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 - 5.70iT - 29T^{2}$$
31 $$1 + 7.23iT - 31T^{2}$$
37 $$1 - 5.23T + 37T^{2}$$
41 $$1 + 6.47T + 41T^{2}$$
43 $$1 + 12.9T + 43T^{2}$$
47 $$1 - 2.47T + 47T^{2}$$
53 $$1 + 8.47iT - 53T^{2}$$
59 $$1 + 4.47T + 59T^{2}$$
61 $$1 + 2.76iT - 61T^{2}$$
67 $$1 - 12T + 67T^{2}$$
71 $$1 - 2.76iT - 71T^{2}$$
73 $$1 - 6.76iT - 73T^{2}$$
79 $$1 + 8.94T + 79T^{2}$$
83 $$1 + 16.6T + 83T^{2}$$
89 $$1 - 14.4T + 89T^{2}$$
97 $$1 + 5.23iT - 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

−9.798878203925078526575674649705, −9.056576299346222354173028169924, −8.280250120189673961002885068455, −7.61708810754020303668367782385, −6.56106077076978569060464554775, −5.39246272762699435325855789867, −4.93131617687550189457514548608, −3.86167420635426188678849682444, −3.00524266907389298534696270949, −2.01706663257402181522387983089, 0.33422167430254205903067072716, 1.49463270926863549536525469824, 2.88987444622136810356448536851, 3.68074304791265306997459786622, 4.63762755889374355324945374256, 5.98279254890133807172330459390, 6.66498553078278138667165030181, 7.26969712866685221702542788923, 8.257727202885729411436940199168, 8.581970573275890258902804584812