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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.58·5-s + 7-s − 4.04·11-s − 0.190·13-s − 6.82·17-s − 1.34·19-s + 0.240·23-s + 1.65·25-s + 0.190·29-s + 2.46·31-s + 2.58·35-s + 10.4·37-s − 2.77·41-s − 11.4·43-s + 4.29·47-s + 49-s + 9.26·53-s − 10.4·55-s − 12.9·59-s − 13.6·61-s − 0.490·65-s − 0.871·67-s − 11.0·71-s + 12.5·73-s − 4.04·77-s − 15.0·79-s − 4.19·83-s + ⋯
L(s)  = 1  + 1.15·5-s + 0.377·7-s − 1.22·11-s − 0.0527·13-s − 1.65·17-s − 0.307·19-s + 0.0500·23-s + 0.331·25-s + 0.0353·29-s + 0.443·31-s + 0.436·35-s + 1.71·37-s − 0.432·41-s − 1.74·43-s + 0.625·47-s + 0.142·49-s + 1.27·53-s − 1.40·55-s − 1.68·59-s − 1.74·61-s − 0.0608·65-s − 0.106·67-s − 1.31·71-s + 1.46·73-s − 0.461·77-s − 1.69·79-s − 0.459·83-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 \( 1 - 2.58T + 5T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + 0.190T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 1.34T + 19T^{2} \)
23 \( 1 - 0.240T + 23T^{2} \)
29 \( 1 - 0.190T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 4.29T + 47T^{2} \)
53 \( 1 - 9.26T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 0.871T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 4.19T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76954723474778312947211965505, −6.94226869461527589511604086296, −6.21184848298912722960328347088, −5.64517033927695183228115278100, −4.82476686547887270950514035331, −4.29547302694165690376662993102, −2.90652485638411626486290873628, −2.33724498217187129317758827266, −1.52975403692924058925749713638, 0, 1.52975403692924058925749713638, 2.33724498217187129317758827266, 2.90652485638411626486290873628, 4.29547302694165690376662993102, 4.82476686547887270950514035331, 5.64517033927695183228115278100, 6.21184848298912722960328347088, 6.94226869461527589511604086296, 7.76954723474778312947211965505

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