Properties

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

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line