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  + 0.314·5-s − 7-s + 1.47·11-s − 1.79·13-s + 5.22·17-s − 6.22·19-s + 6.58·23-s − 4.90·25-s − 7.43·29-s + 4.43·31-s − 0.314·35-s − 2.79·37-s − 8.37·41-s + 0.207·43-s − 4.06·47-s + 49-s − 4·53-s + 0.464·55-s − 4.06·59-s − 1.67·61-s − 0.563·65-s + 13.9·67-s − 7.27·71-s + 14.6·73-s − 1.47·77-s − 1.46·79-s + 12.2·83-s + ⋯
L(s)  = 1  + 0.140·5-s − 0.377·7-s + 0.445·11-s − 0.497·13-s + 1.26·17-s − 1.42·19-s + 1.37·23-s − 0.980·25-s − 1.38·29-s + 0.796·31-s − 0.0531·35-s − 0.459·37-s − 1.30·41-s + 0.0316·43-s − 0.592·47-s + 0.142·49-s − 0.549·53-s + 0.0626·55-s − 0.529·59-s − 0.214·61-s − 0.0698·65-s + 1.70·67-s − 0.862·71-s + 1.71·73-s − 0.168·77-s − 0.164·79-s + 1.34·83-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.314T + 5T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + 1.79T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 + 7.43T + 29T^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 + 2.79T + 37T^{2} \)
41 \( 1 + 8.37T + 41T^{2} \)
43 \( 1 - 0.207T + 43T^{2} \)
47 \( 1 + 4.06T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 4.06T + 59T^{2} \)
61 \( 1 + 1.67T + 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 + 7.27T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−7.77734019633866849076314697281, −6.89173389821059298759210361504, −6.42286602092850047792319428607, −5.53917169568997885875609847592, −4.92414424898903618862685177665, −3.92923182619190741137687358340, −3.30130433968494897717015539923, −2.31055093673384561781893488543, −1.36033027197864957309686515364, 0, 1.36033027197864957309686515364, 2.31055093673384561781893488543, 3.30130433968494897717015539923, 3.92923182619190741137687358340, 4.92414424898903618862685177665, 5.53917169568997885875609847592, 6.42286602092850047792319428607, 6.89173389821059298759210361504, 7.77734019633866849076314697281

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