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  + 2.83·5-s − 7-s + 2.47·11-s − 5.31·13-s − 5.78·17-s + 4.78·19-s − 3.90·23-s + 3.06·25-s + 7.10·29-s − 10.1·31-s − 2.83·35-s − 6.31·37-s − 1.41·41-s − 3.31·43-s + 5.42·47-s + 49-s − 4·53-s + 7.03·55-s + 5.42·59-s − 4.72·61-s − 15.1·65-s − 1.52·67-s − 11.7·71-s − 10.8·73-s − 2.47·77-s − 8.03·79-s + 14.8·83-s + ⋯
L(s)  = 1  + 1.26·5-s − 0.377·7-s + 0.747·11-s − 1.47·13-s − 1.40·17-s + 1.09·19-s − 0.813·23-s + 0.612·25-s + 1.31·29-s − 1.81·31-s − 0.479·35-s − 1.03·37-s − 0.221·41-s − 0.506·43-s + 0.790·47-s + 0.142·49-s − 0.549·53-s + 0.949·55-s + 0.705·59-s − 0.604·61-s − 1.87·65-s − 0.186·67-s − 1.40·71-s − 1.27·73-s − 0.282·77-s − 0.904·79-s + 1.63·83-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 \( 1 - 2.83T + 5T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 - 4.78T + 19T^{2} \)
23 \( 1 + 3.90T + 23T^{2} \)
29 \( 1 - 7.10T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 6.31T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 3.31T + 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 5.42T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 8.03T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + 5.41T + 89T^{2} \)
97 \( 1 + 2.84T + 97T^{2} \)
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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

−7.50607165313917817405821921110, −6.95952019156304978907944493085, −6.32408189249797148068565732060, −5.59565222499160576465009108261, −4.94445792799587683605361185019, −4.11395320790515154638683138406, −3.05693674830878706285170586134, −2.24772974770745639588161047866, −1.54253869646833641274836028975, 0, 1.54253869646833641274836028975, 2.24772974770745639588161047866, 3.05693674830878706285170586134, 4.11395320790515154638683138406, 4.94445792799587683605361185019, 5.59565222499160576465009108261, 6.32408189249797148068565732060, 6.95952019156304978907944493085, 7.50607165313917817405821921110

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