# Properties

 Degree $2$ Conductor $3840$ Sign $0.316 - 0.948i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3-s + (−1 − 2i)5-s + 2i·7-s + 9-s + 2i·11-s + 6·13-s + (1 + 2i)15-s − 2i·17-s − 2i·21-s + 4i·23-s + (−3 + 4i)25-s − 27-s − 8·31-s − 2i·33-s + (4 − 2i)35-s + ⋯
 L(s)  = 1 − 0.577·3-s + (−0.447 − 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 0.603i·11-s + 1.66·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s − 0.436i·21-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s − 1.43·31-s − 0.348i·33-s + (0.676 − 0.338i)35-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3840$$    =    $$2^{8} \cdot 3 \cdot 5$$ Sign: $0.316 - 0.948i$ Motivic weight: $$1$$ Character: $\chi_{3840} (2689, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3840,\ (\ :1/2),\ 0.316 - 0.948i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.112749778$$ $$L(\frac12)$$ $$\approx$$ $$1.112749778$$ $$L(\frac{3}{2})$$ 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 + T$$
5 $$1 + (1 + 2i)T$$
good7 $$1 - 2iT - 7T^{2}$$
11 $$1 - 2iT - 11T^{2}$$
13 $$1 - 6T + 13T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 8T + 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 - 8iT - 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + 10iT - 59T^{2}$$
61 $$1 + 2iT - 61T^{2}$$
67 $$1 - 8T + 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 4T + 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 - 8iT - 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

−8.734379806350938919729633373185, −7.908853449102611121118775583871, −7.23987004492467640026617087566, −6.22258883538268339401687359678, −5.62852528967901714807388011888, −4.97146855756344697124598267090, −4.11548539221463832232546862790, −3.35154896961385924792478773281, −1.93756372283623915499700052105, −1.03975617128410289720254036218, 0.42137465715308899645928942561, 1.60187568340872147871724595077, 2.99222608237856889134273527132, 3.82186515933564960511372617941, 4.25972264733339860129855668421, 5.60358478788117515773885525188, 6.14007842572282967512468526645, 6.84720043549624350358626348772, 7.46483876337736422533928003126, 8.325806178665803861551864609891