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  − 7-s + 4·11-s + 13-s − 5·17-s + 2·19-s − 3·23-s − 5·25-s + 29-s − 9·31-s + 6·37-s + 2·41-s + 43-s + 2·47-s + 49-s − 9·53-s + 3·59-s − 2·61-s − 11·67-s − 5·71-s − 2·73-s − 4·77-s − 14·79-s − 12·83-s + 15·89-s − 91-s + 12·97-s + 6·101-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 0.277·13-s − 1.21·17-s + 0.458·19-s − 0.625·23-s − 25-s + 0.185·29-s − 1.61·31-s + 0.986·37-s + 0.312·41-s + 0.152·43-s + 0.291·47-s + 1/7·49-s − 1.23·53-s + 0.390·59-s − 0.256·61-s − 1.34·67-s − 0.593·71-s − 0.234·73-s − 0.455·77-s − 1.57·79-s − 1.31·83-s + 1.58·89-s − 0.104·91-s + 1.21·97-s + 0.597·101-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.59311188580110539363116794660, −7.05871557701444405382575641058, −6.14206798720315933572774310078, −5.88802297225246102590812321359, −4.65567221057861950951876745078, −4.03695149909066385948072829856, −3.34643719282074331294912065543, −2.25368023021477087787970864433, −1.37807743210823378160344702809, 0, 1.37807743210823378160344702809, 2.25368023021477087787970864433, 3.34643719282074331294912065543, 4.03695149909066385948072829856, 4.65567221057861950951876745078, 5.88802297225246102590812321359, 6.14206798720315933572774310078, 7.05871557701444405382575641058, 7.59311188580110539363116794660

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