Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.5 + 0.866i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46i·5-s − 2.64·7-s − 4.58i·13-s + 1.73i·17-s − 5.29·19-s − 4.58i·23-s − 6.99·25-s + 7.93·29-s − 2.64·31-s − 9.16i·35-s + 4·37-s − 6.92i·41-s + 1.73i·43-s − 6·47-s + 7.00·49-s + ⋯
L(s)  = 1  + 1.54i·5-s − 0.999·7-s − 1.27i·13-s + 0.420i·17-s − 1.21·19-s − 0.955i·23-s − 1.39·25-s + 1.47·29-s − 0.475·31-s − 1.54i·35-s + 0.657·37-s − 1.08i·41-s + 0.264i·43-s − 0.875·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.5 + 0.866i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.5 + 0.866i)\)
\(L(1)\)  \(\approx\)  \(0.8776394971\)
\(L(\frac12)\)  \(\approx\)  \(0.8776394971\)
\(L(\frac{3}{2})\)   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 \( 1 + 2.64T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4.58iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 9.16iT - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 4.58iT - 71T^{2} \)
73 \( 1 - 9.16iT - 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 5.19iT - 89T^{2} \)
97 \( 1 - 9.16iT - 97T^{2} \)
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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

−8.455098356724067382512696698516, −7.84670087808327420889141006231, −6.86683169056934973055324779701, −6.42716797466483306966862560216, −5.87370656747971866759144125921, −4.63215516681622433331001492695, −3.54581441697250822623071516613, −2.99692983859679776130038916266, −2.21486702787989740790102378284, −0.30996924983463594574558091552, 1.02634958463653788423674866876, 2.10826812791739662003575078402, 3.33700483409131087015183657450, 4.38317100606364102124423469455, 4.76335103094274571025763720278, 5.88057113563610627400037320937, 6.47870380945965778313310664847, 7.33935125088105743776056311064, 8.335623516144387085833434931604, 8.879374791507878557709670929902

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