Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.755 - 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·5-s + (−0.5 + 2.59i)7-s + 5.19i·11-s + 3.46i·13-s + 3.46i·17-s + 8·19-s − 6.92i·23-s + 2.00·25-s + 6·29-s − 5·31-s + (−4.5 − 0.866i)35-s + 4·37-s + 6.92i·41-s + 3.46i·43-s − 6·47-s + ⋯
L(s)  = 1  + 0.774i·5-s + (−0.188 + 0.981i)7-s + 1.56i·11-s + 0.960i·13-s + 0.840i·17-s + 1.83·19-s − 1.44i·23-s + 0.400·25-s + 1.11·29-s − 0.898·31-s + (−0.760 − 0.146i)35-s + 0.657·37-s + 1.08i·41-s + 0.528i·43-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.755 - 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.755 - 0.654i)\)
\(L(1)\)  \(\approx\)  \(1.698479425\)
\(L(\frac12)\)  \(\approx\)  \(1.698479425\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 3.46iT - 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−9.154480413383507872776540143350, −8.230659355500302233329943368466, −7.44158493876838305652500989366, −6.64736846153915600103018875532, −6.23791272768971002080021076306, −5.05005180770035736117198639534, −4.47239690492159970291767306484, −3.26290870978848290754273136398, −2.50456312538850418832409855782, −1.58946466492516623393880001500, 0.60253261047836479686556714910, 1.20458629331792589701431308306, 3.09037940669844404205461566467, 3.41458956676008800958214186861, 4.62065043187181382590848178771, 5.43383286617479293384194166205, 5.90972994336019980328210711859, 7.22293875315507179668820537662, 7.56147787022240455212107554339, 8.485748344008287363452887878158

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