Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·5-s + 7-s − 2.41·11-s + 2.82·13-s − 2.82·17-s + 3.82·19-s − 5.24·23-s + 0.828·25-s + 2.82·29-s + 4.65·31-s − 2.41·35-s + 0.171·37-s + 2.07·41-s + 4.82·43-s − 0.343·47-s + 49-s + 4·53-s + 5.82·55-s + 3.65·59-s − 11.3·61-s − 6.82·65-s − 4.48·67-s − 5.58·71-s − 3.17·73-s − 2.41·77-s − 4.82·79-s − 10.8·83-s + ⋯
L(s)  = 1  − 1.07·5-s + 0.377·7-s − 0.727·11-s + 0.784·13-s − 0.685·17-s + 0.878·19-s − 1.09·23-s + 0.165·25-s + 0.525·29-s + 0.836·31-s − 0.408·35-s + 0.0282·37-s + 0.323·41-s + 0.736·43-s − 0.0500·47-s + 0.142·49-s + 0.549·53-s + 0.785·55-s + 0.476·59-s − 1.44·61-s − 0.846·65-s − 0.547·67-s − 0.662·71-s − 0.371·73-s − 0.275·77-s − 0.543·79-s − 1.18·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 2.41T + 5T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 - 0.171T + 37T^{2} \)
41 \( 1 - 2.07T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 + 5.58T + 71T^{2} \)
73 \( 1 + 3.17T + 73T^{2} \)
79 \( 1 + 4.82T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 1.24T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−7.72911258868049666759784880923, −7.25310505158434878971577178009, −6.25496498432210115765214637179, −5.61630816649468575215826929470, −4.62573260263317666763402331371, −4.14213487929080329600589400748, −3.28005564770321649212792263576, −2.41963106878474376545201733131, −1.20602011349772668883187851424, 0, 1.20602011349772668883187851424, 2.41963106878474376545201733131, 3.28005564770321649212792263576, 4.14213487929080329600589400748, 4.62573260263317666763402331371, 5.61630816649468575215826929470, 6.25496498432210115765214637179, 7.25310505158434878971577178009, 7.72911258868049666759784880923

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