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  + 3·5-s + 7-s − 2·11-s − 4·13-s − 5·17-s + 4·23-s + 4·25-s − 4·29-s − 8·31-s + 3·35-s + 37-s − 7·41-s + 5·43-s − 47-s + 49-s − 2·53-s − 6·55-s + 11·59-s − 14·61-s − 12·65-s − 4·67-s − 12·71-s − 2·77-s + 13·79-s − 11·83-s − 15·85-s − 6·89-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s − 1.21·17-s + 0.834·23-s + 4/5·25-s − 0.742·29-s − 1.43·31-s + 0.507·35-s + 0.164·37-s − 1.09·41-s + 0.762·43-s − 0.145·47-s + 1/7·49-s − 0.274·53-s − 0.809·55-s + 1.43·59-s − 1.79·61-s − 1.48·65-s − 0.488·67-s − 1.42·71-s − 0.227·77-s + 1.46·79-s − 1.20·83-s − 1.62·85-s − 0.635·89-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  :  $\chi_{6048} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−17.72178377919464, −17.17421884435847, −16.73286069258062, −16.00356747951113, −15.06808850437684, −14.86768728202910, −14.07195358469662, −13.47138051028892, −13.02827687734975, −12.47422132450582, −11.57763901481678, −10.88727532881815, −10.42646178729278, −9.640168266475678, −9.197174771536666, −8.565489996516474, −7.559146986578695, −7.067535707304907, −6.252940274288318, −5.449606713482773, −5.048139708426062, −4.197636287407084, −2.957756523274450, −2.241708672213603, −1.595805183569122, 0, 1.595805183569122, 2.241708672213603, 2.957756523274450, 4.197636287407084, 5.048139708426062, 5.449606713482773, 6.252940274288318, 7.067535707304907, 7.559146986578695, 8.565489996516474, 9.197174771536666, 9.640168266475678, 10.42646178729278, 10.88727532881815, 11.57763901481678, 12.47422132450582, 13.02827687734975, 13.47138051028892, 14.07195358469662, 14.86768728202910, 15.06808850437684, 16.00356747951113, 16.73286069258062, 17.17421884435847, 17.72178377919464

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