Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.377 - 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.377 - 0.925i)\)
\(L(1)\)  \(\approx\)  \(1.860987433\)
\(L(\frac12)\)  \(\approx\)  \(1.860987433\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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