Properties

 Degree $2$ Conductor $3024$ Sign $-0.377 + 0.925i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 1.73·5-s + (1 − 2.44i)7-s − 4.24i·11-s − 2.44i·13-s + 1.73·17-s − 2.44i·19-s − 8.48i·23-s − 2.00·25-s + 4.24i·29-s + 7.34i·31-s + (1.73 − 4.24i)35-s + 37-s − 1.73·41-s − 7·43-s − 12.1·47-s + ⋯
 L(s)  = 1 + 0.774·5-s + (0.377 − 0.925i)7-s − 1.27i·11-s − 0.679i·13-s + 0.420·17-s − 0.561i·19-s − 1.76i·23-s − 0.400·25-s + 0.787i·29-s + 1.31i·31-s + (0.292 − 0.717i)35-s + 0.164·37-s − 0.270·41-s − 1.06·43-s − 1.76·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $-0.377 + 0.925i$ Motivic weight: $$1$$ Character: $\chi_{3024} (1889, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :1/2),\ -0.377 + 0.925i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.860987433$$ $$L(\frac12)$$ $$\approx$$ $$1.860987433$$ $$L(\frac{3}{2})$$ 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 + (-1 + 2.44i)T$$
good5 $$1 - 1.73T + 5T^{2}$$
11 $$1 + 4.24iT - 11T^{2}$$
13 $$1 + 2.44iT - 13T^{2}$$
17 $$1 - 1.73T + 17T^{2}$$
19 $$1 + 2.44iT - 19T^{2}$$
23 $$1 + 8.48iT - 23T^{2}$$
29 $$1 - 4.24iT - 29T^{2}$$
31 $$1 - 7.34iT - 31T^{2}$$
37 $$1 - T + 37T^{2}$$
41 $$1 + 1.73T + 41T^{2}$$
43 $$1 + 7T + 43T^{2}$$
47 $$1 + 12.1T + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 8.66T + 59T^{2}$$
61 $$1 - 2.44iT - 61T^{2}$$
67 $$1 - 10T + 67T^{2}$$
71 $$1 + 8.48iT - 71T^{2}$$
73 $$1 - 9.79iT - 73T^{2}$$
79 $$1 + 5T + 79T^{2}$$
83 $$1 - 12.1T + 83T^{2}$$
89 $$1 - 10.3T + 89T^{2}$$
97 $$1 + 2.44iT - 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

−8.393429511180140727998793496278, −7.924585570759252383672264222530, −6.79578460914676005506353531107, −6.34299734581220373938676842747, −5.33164673248896392677464896767, −4.77448678283064511973801215807, −3.57869965554652195770290364262, −2.88737819931145486457547640673, −1.59709802105164168344297905258, −0.55503927228159515059183686625, 1.75770427776262166650486565344, 2.01208597523022734434091820916, 3.32432077125894175018974792028, 4.39680392659725505441431372008, 5.21012889090947761970081452266, 5.87058944076916705566913397397, 6.57124236584979775426725021075, 7.59772322587017670794869649115, 8.093035265933463506337022431160, 9.179670298996030349200139633682