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  − 4.38·5-s − 7-s + 3.59·11-s + 0.797·13-s − 0.295·17-s − 0.704·19-s − 7.86·23-s + 14.2·25-s − 4.50·29-s + 1.50·31-s + 4.38·35-s − 0.202·37-s + 8.66·41-s + 2.79·43-s + 8.27·47-s + 49-s − 4·53-s − 15.7·55-s + 8.27·59-s + 11.9·61-s − 3.49·65-s − 13.3·67-s − 6.79·71-s + 6.20·73-s − 3.59·77-s + 14.7·79-s − 14.6·83-s + ⋯
L(s)  = 1  − 1.96·5-s − 0.377·7-s + 1.08·11-s + 0.221·13-s − 0.0717·17-s − 0.161·19-s − 1.63·23-s + 2.85·25-s − 0.835·29-s + 0.269·31-s + 0.741·35-s − 0.0333·37-s + 1.35·41-s + 0.426·43-s + 1.20·47-s + 0.142·49-s − 0.549·53-s − 2.12·55-s + 1.07·59-s + 1.53·61-s − 0.433·65-s − 1.63·67-s − 0.806·71-s + 0.726·73-s − 0.409·77-s + 1.65·79-s − 1.60·83-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 \( 1 + 4.38T + 5T^{2} \)
11 \( 1 - 3.59T + 11T^{2} \)
13 \( 1 - 0.797T + 13T^{2} \)
17 \( 1 + 0.295T + 17T^{2} \)
19 \( 1 + 0.704T + 19T^{2} \)
23 \( 1 + 7.86T + 23T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + 0.202T + 37T^{2} \)
41 \( 1 - 8.66T + 41T^{2} \)
43 \( 1 - 2.79T + 43T^{2} \)
47 \( 1 - 8.27T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 8.27T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 - 6.20T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 4.66T + 89T^{2} \)
97 \( 1 + 8.54T + 97T^{2} \)
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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

−7.70876623753057814384903183284, −7.14455976403401319273501880984, −6.43303498322178400357613532970, −5.65038130369142942981836975726, −4.45167745989121934425277728738, −3.98324755427311215372226591526, −3.55505141096117627634723793815, −2.45722446900485592197474343902, −1.05911064826559099409753438687, 0, 1.05911064826559099409753438687, 2.45722446900485592197474343902, 3.55505141096117627634723793815, 3.98324755427311215372226591526, 4.45167745989121934425277728738, 5.65038130369142942981836975726, 6.43303498322178400357613532970, 7.14455976403401319273501880984, 7.70876623753057814384903183284

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