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  − 1.34·5-s − 7-s + 6.28·11-s − 4.43·13-s − 0.505·17-s + 6.19·19-s − 5.84·23-s − 3.19·25-s − 4.43·29-s + 6.62·31-s + 1.34·35-s − 8.44·37-s + 5.77·41-s − 7.44·43-s − 12.1·47-s + 49-s + 13.8·53-s − 8.44·55-s − 6.25·59-s + 1.01·61-s + 5.95·65-s + 14.8·67-s + 2.35·71-s + 11.8·73-s − 6.28·77-s − 12.3·79-s − 8.43·83-s + ⋯
L(s)  = 1  − 0.601·5-s − 0.377·7-s + 1.89·11-s − 1.22·13-s − 0.122·17-s + 1.42·19-s − 1.21·23-s − 0.638·25-s − 0.822·29-s + 1.18·31-s + 0.227·35-s − 1.38·37-s + 0.901·41-s − 1.13·43-s − 1.76·47-s + 0.142·49-s + 1.90·53-s − 1.13·55-s − 0.814·59-s + 0.129·61-s + 0.738·65-s + 1.81·67-s + 0.279·71-s + 1.38·73-s − 0.715·77-s − 1.38·79-s − 0.925·83-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 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 - 6.28T + 11T^{2} \)
13 \( 1 + 4.43T + 13T^{2} \)
17 \( 1 + 0.505T + 17T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 + 5.84T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 - 6.62T + 31T^{2} \)
37 \( 1 + 8.44T + 37T^{2} \)
41 \( 1 - 5.77T + 41T^{2} \)
43 \( 1 + 7.44T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 6.25T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 - 2.35T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 8.43T + 83T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−7.66712135169256230398566026159, −6.99198001871856476008448086021, −6.46871736054527338450853546356, −5.57816072287223510890159894637, −4.75901140128448480615555119025, −3.86844915913480228476422684241, −3.48642541133279725901311015373, −2.30952654347509147752282504039, −1.27885108379442852892982304539, 0, 1.27885108379442852892982304539, 2.30952654347509147752282504039, 3.48642541133279725901311015373, 3.86844915913480228476422684241, 4.75901140128448480615555119025, 5.57816072287223510890159894637, 6.46871736054527338450853546356, 6.99198001871856476008448086021, 7.66712135169256230398566026159

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