Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.29·5-s + 7-s + 5.01·11-s − 4.55·13-s + 0.762·17-s − 3.01·19-s − 1.53·23-s + 0.278·25-s + 3.29·29-s + 3.29·31-s + 2.29·35-s + 4.72·37-s + 0.297·41-s − 3.83·43-s + 5.79·47-s + 49-s + 11.3·53-s + 11.5·55-s − 3.59·59-s + 2.42·61-s − 10.4·65-s + 4.12·67-s − 2.46·71-s + 16.6·73-s + 5.01·77-s − 5.36·79-s + 1.27·83-s + ⋯
L(s)  = 1  + 1.02·5-s + 0.377·7-s + 1.51·11-s − 1.26·13-s + 0.185·17-s − 0.692·19-s − 0.319·23-s + 0.0557·25-s + 0.612·29-s + 0.592·31-s + 0.388·35-s + 0.776·37-s + 0.0464·41-s − 0.584·43-s + 0.844·47-s + 0.142·49-s + 1.56·53-s + 1.55·55-s − 0.468·59-s + 0.310·61-s − 1.29·65-s + 0.504·67-s − 0.292·71-s + 1.94·73-s + 0.571·77-s − 0.603·79-s + 0.140·83-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  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.758973535$
$L(\frac12)$  $\approx$  $2.758973535$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 5T^{2} \)
11 \( 1 - 5.01T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 - 0.762T + 17T^{2} \)
19 \( 1 + 3.01T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 - 3.29T + 29T^{2} \)
31 \( 1 - 3.29T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 - 0.297T + 41T^{2} \)
43 \( 1 + 3.83T + 43T^{2} \)
47 \( 1 - 5.79T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 3.59T + 59T^{2} \)
61 \( 1 - 2.42T + 61T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 + 2.46T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 + 5.36T + 79T^{2} \)
83 \( 1 - 1.27T + 83T^{2} \)
89 \( 1 - 9.57T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.098626274181228877500685812388, −7.29033390886719756649354604540, −6.53644555125212245384129116291, −6.06879085085583240541273358106, −5.20022366758776035691150117338, −4.49510082745815113844510194625, −3.73603440455246301396918772860, −2.52333867345252177263824068008, −1.94372461600315813315499762559, −0.899808095297293235638379748060, 0.899808095297293235638379748060, 1.94372461600315813315499762559, 2.52333867345252177263824068008, 3.73603440455246301396918772860, 4.49510082745815113844510194625, 5.20022366758776035691150117338, 6.06879085085583240541273358106, 6.53644555125212245384129116291, 7.29033390886719756649354604540, 8.098626274181228877500685812388

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