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  + 2.48·2-s + 4.15·4-s + 0.783·5-s + 7-s + 5.33·8-s + 1.94·10-s − 4.88·11-s − 6.65·13-s + 2.48·14-s + 4.92·16-s − 1.25·17-s − 2.07·19-s + 3.25·20-s − 12.1·22-s − 6.86·23-s − 4.38·25-s − 16.5·26-s + 4.15·28-s + 9.79·29-s − 3.45·31-s + 1.54·32-s − 3.11·34-s + 0.783·35-s − 2.67·37-s − 5.15·38-s + 4.18·40-s + 4.19·41-s + ⋯
L(s)  = 1  + 1.75·2-s + 2.07·4-s + 0.350·5-s + 0.377·7-s + 1.88·8-s + 0.614·10-s − 1.47·11-s − 1.84·13-s + 0.662·14-s + 1.23·16-s − 0.304·17-s − 0.476·19-s + 0.727·20-s − 2.58·22-s − 1.43·23-s − 0.877·25-s − 3.23·26-s + 0.784·28-s + 1.81·29-s − 0.620·31-s + 0.273·32-s − 0.534·34-s + 0.132·35-s − 0.440·37-s − 0.836·38-s + 0.661·40-s + 0.655·41-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 \( 1 - 2.48T + 2T^{2} \)
5 \( 1 - 0.783T + 5T^{2} \)
11 \( 1 + 4.88T + 11T^{2} \)
13 \( 1 + 6.65T + 13T^{2} \)
17 \( 1 + 1.25T + 17T^{2} \)
19 \( 1 + 2.07T + 19T^{2} \)
23 \( 1 + 6.86T + 23T^{2} \)
29 \( 1 - 9.79T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 - 6.51T + 47T^{2} \)
53 \( 1 + 3.48T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 6.13T + 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 - 7.18T + 71T^{2} \)
73 \( 1 - 2.24T + 73T^{2} \)
79 \( 1 + 7.38T + 79T^{2} \)
83 \( 1 - 3.82T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−7.55327392326512782936242486037, −6.47534270440314337760615077845, −5.93535205523418570756006285199, −5.16714225064817205098379018272, −4.78082629807608512392922377367, −4.15872766151730334792945709437, −3.09937233980481292456569367148, −2.36971574895449476526623964965, −2.00482167194477589797158999486, 0, 2.00482167194477589797158999486, 2.36971574895449476526623964965, 3.09937233980481292456569367148, 4.15872766151730334792945709437, 4.78082629807608512392922377367, 5.16714225064817205098379018272, 5.93535205523418570756006285199, 6.47534270440314337760615077845, 7.55327392326512782936242486037

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