Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
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  − 0.242·2-s − 1.94·4-s + 2.79·5-s − 7-s + 0.955·8-s − 0.678·10-s + 4.59·11-s − 4.92·13-s + 0.242·14-s + 3.65·16-s − 2.21·17-s + 0.299·19-s − 5.42·20-s − 1.11·22-s − 2.16·23-s + 2.82·25-s + 1.19·26-s + 1.94·28-s − 5.57·29-s + 6.00·31-s − 2.79·32-s + 0.536·34-s − 2.79·35-s + 8.81·37-s − 0.0726·38-s + 2.67·40-s − 10.5·41-s + ⋯
L(s)  = 1  − 0.171·2-s − 0.970·4-s + 1.25·5-s − 0.377·7-s + 0.337·8-s − 0.214·10-s + 1.38·11-s − 1.36·13-s + 0.0648·14-s + 0.912·16-s − 0.536·17-s + 0.0687·19-s − 1.21·20-s − 0.237·22-s − 0.452·23-s + 0.564·25-s + 0.234·26-s + 0.366·28-s − 1.03·29-s + 1.07·31-s − 0.494·32-s + 0.0920·34-s − 0.472·35-s + 1.44·37-s − 0.0117·38-s + 0.422·40-s − 1.64·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 + T \)
127 \( 1 + T \)
good2 \( 1 + 0.242T + 2T^{2} \)
5 \( 1 - 2.79T + 5T^{2} \)
11 \( 1 - 4.59T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 + 2.21T + 17T^{2} \)
19 \( 1 - 0.299T + 19T^{2} \)
23 \( 1 + 2.16T + 23T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 - 6.00T + 31T^{2} \)
37 \( 1 - 8.81T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 6.69T + 43T^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 + 1.04T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 - 0.161T + 61T^{2} \)
67 \( 1 - 2.60T + 67T^{2} \)
71 \( 1 + 2.49T + 71T^{2} \)
73 \( 1 + 5.94T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 - 9.33T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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

−7.45841985777973452252176519845, −6.72119657941708862332902582751, −6.07892488386335115460581879221, −5.43950626925740967005999908772, −4.63469782870010935962206301964, −4.07360946456560379465743366615, −3.06643282773240484468104034297, −2.10321870705675055431920552204, −1.27927944899865137883141531782, 0, 1.27927944899865137883141531782, 2.10321870705675055431920552204, 3.06643282773240484468104034297, 4.07360946456560379465743366615, 4.63469782870010935962206301964, 5.43950626925740967005999908772, 6.07892488386335115460581879221, 6.72119657941708862332902582751, 7.45841985777973452252176519845

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