# Properties

 Degree $2$ Conductor $6048$ Sign $0.826 + 0.563i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.52i·5-s − 7-s + 5.70i·11-s − 3.06i·13-s + 5.49·17-s − 7.28i·19-s + 0.539·23-s − 1.38·25-s − 8.35i·29-s − 6.74·31-s − 2.52i·35-s − 10.2i·37-s − 5.58·41-s − 3.98i·43-s + 4.83·47-s + ⋯
 L(s)  = 1 + 1.12i·5-s − 0.377·7-s + 1.71i·11-s − 0.848i·13-s + 1.33·17-s − 1.67i·19-s + 0.112·23-s − 0.276·25-s − 1.55i·29-s − 1.21·31-s − 0.427i·35-s − 1.68i·37-s − 0.872·41-s − 0.607i·43-s + 0.705·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ Sign: $0.826 + 0.563i$ Motivic weight: $$1$$ Character: $\chi_{6048} (3025, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 6048,\ (\ :1/2),\ 0.826 + 0.563i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.516464153$$ $$L(\frac12)$$ $$\approx$$ $$1.516464153$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 + T$$
good5 $$1 - 2.52iT - 5T^{2}$$
11 $$1 - 5.70iT - 11T^{2}$$
13 $$1 + 3.06iT - 13T^{2}$$
17 $$1 - 5.49T + 17T^{2}$$
19 $$1 + 7.28iT - 19T^{2}$$
23 $$1 - 0.539T + 23T^{2}$$
29 $$1 + 8.35iT - 29T^{2}$$
31 $$1 + 6.74T + 31T^{2}$$
37 $$1 + 10.2iT - 37T^{2}$$
41 $$1 + 5.58T + 41T^{2}$$
43 $$1 + 3.98iT - 43T^{2}$$
47 $$1 - 4.83T + 47T^{2}$$
53 $$1 + 11.1iT - 53T^{2}$$
59 $$1 - 10.1iT - 59T^{2}$$
61 $$1 + 4.14iT - 61T^{2}$$
67 $$1 - 5.96iT - 67T^{2}$$
71 $$1 + 4.93T + 71T^{2}$$
73 $$1 - 8.66T + 73T^{2}$$
79 $$1 + 12.0T + 79T^{2}$$
83 $$1 + 5.83iT - 83T^{2}$$
89 $$1 + 9.28T + 89T^{2}$$
97 $$1 - 12.3T + 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

−7.66749769681163317033624574066, −7.25318655589556196804863220682, −6.85487355124074903773365994512, −5.85575035759980581407251579116, −5.23381938785630537591751684229, −4.28915882766614765970098036795, −3.45688891452991902737456556585, −2.68715171124867381224553322027, −2.00463264809297039919460520844, −0.44197698104252919093145469429, 1.00884246225050178491521901746, 1.60261862329206593572192102879, 3.20166362894334016700763578369, 3.53409010237672926878073241785, 4.56451931049401484866674843430, 5.43297954673078668075550185149, 5.82850833331372921753789458913, 6.62655387647608498711922819375, 7.58477099085768736622942250885, 8.306996214517805362167609838171