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  − 0.246·2-s − 1.93·4-s + 0.318·5-s + 7-s + 0.969·8-s − 0.0783·10-s − 2.79·11-s − 6.49·13-s − 0.246·14-s + 3.63·16-s + 1.33·17-s + 4.36·19-s − 0.617·20-s + 0.689·22-s + 5.29·23-s − 4.89·25-s + 1.60·26-s − 1.93·28-s + 2.35·29-s − 4.76·31-s − 2.83·32-s − 0.327·34-s + 0.318·35-s + 1.06·37-s − 1.07·38-s + 0.308·40-s + 1.46·41-s + ⋯
L(s)  = 1  − 0.174·2-s − 0.969·4-s + 0.142·5-s + 0.377·7-s + 0.342·8-s − 0.0247·10-s − 0.844·11-s − 1.80·13-s − 0.0658·14-s + 0.909·16-s + 0.322·17-s + 1.00·19-s − 0.137·20-s + 0.146·22-s + 1.10·23-s − 0.979·25-s + 0.313·26-s − 0.366·28-s + 0.436·29-s − 0.855·31-s − 0.501·32-s − 0.0562·34-s + 0.0537·35-s + 0.175·37-s − 0.174·38-s + 0.0487·40-s + 0.228·41-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 + 0.246T + 2T^{2} \)
5 \( 1 - 0.318T + 5T^{2} \)
11 \( 1 + 2.79T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 - 1.06T + 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 - 8.04T + 43T^{2} \)
47 \( 1 - 7.16T + 47T^{2} \)
53 \( 1 + 9.12T + 53T^{2} \)
59 \( 1 - 5.02T + 59T^{2} \)
61 \( 1 + 2.48T + 61T^{2} \)
67 \( 1 - 0.549T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 1.49T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 7.20T + 97T^{2} \)
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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

−7.60237137958230389342285360965, −7.12297997033867345026264721559, −5.84036423274528347487965983777, −5.22947146953231800353223277007, −4.85840830436234021417521892613, −4.02647375913777223556066178413, −3.03817039343586453247763620269, −2.28353828024938539006382076250, −1.07475042600521424257386195034, 0, 1.07475042600521424257386195034, 2.28353828024938539006382076250, 3.03817039343586453247763620269, 4.02647375913777223556066178413, 4.85840830436234021417521892613, 5.22947146953231800353223277007, 5.84036423274528347487965983777, 7.12297997033867345026264721559, 7.60237137958230389342285360965

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