Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.45·2-s + 0.117·4-s + 0.420·5-s + 7-s + 2.73·8-s − 0.611·10-s − 4.09·11-s + 5.95·13-s − 1.45·14-s − 4.22·16-s + 7.99·17-s + 1.20·19-s + 0.0491·20-s + 5.95·22-s + 0.806·23-s − 4.82·25-s − 8.66·26-s + 0.117·28-s − 5.46·29-s + 3.58·31-s + 0.661·32-s − 11.6·34-s + 0.420·35-s − 3.79·37-s − 1.74·38-s + 1.15·40-s − 9.24·41-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0585·4-s + 0.187·5-s + 0.377·7-s + 0.968·8-s − 0.193·10-s − 1.23·11-s + 1.65·13-s − 0.388·14-s − 1.05·16-s + 1.93·17-s + 0.275·19-s + 0.0109·20-s + 1.26·22-s + 0.168·23-s − 0.964·25-s − 1.70·26-s + 0.0221·28-s − 1.01·29-s + 0.644·31-s + 0.116·32-s − 1.99·34-s + 0.0709·35-s − 0.624·37-s − 0.283·38-s + 0.181·40-s − 1.44·41-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 \( 1 + 1.45T + 2T^{2} \)
5 \( 1 - 0.420T + 5T^{2} \)
11 \( 1 + 4.09T + 11T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
17 \( 1 - 7.99T + 17T^{2} \)
19 \( 1 - 1.20T + 19T^{2} \)
23 \( 1 - 0.806T + 23T^{2} \)
29 \( 1 + 5.46T + 29T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 + 3.79T + 37T^{2} \)
41 \( 1 + 9.24T + 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 3.29T + 53T^{2} \)
59 \( 1 + 5.19T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 8.24T + 71T^{2} \)
73 \( 1 - 8.96T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 8.63T + 97T^{2} \)
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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

−7.79365679320016009508859904653, −7.10057754526753849229089046283, −6.06532484791985065350593650479, −5.42986897838059700889658113270, −4.84114732436266250583224712862, −3.73235321795861931113203605339, −3.13293191972145674260065180862, −1.77340315613627045001134263824, −1.24908626536862198568871970716, 0, 1.24908626536862198568871970716, 1.77340315613627045001134263824, 3.13293191972145674260065180862, 3.73235321795861931113203605339, 4.84114732436266250583224712862, 5.42986897838059700889658113270, 6.06532484791985065350593650479, 7.10057754526753849229089046283, 7.79365679320016009508859904653

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