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  + 2.15·2-s + 2.64·4-s + 0.239·5-s + 7-s + 1.40·8-s + 0.515·10-s + 3.99·11-s − 1.61·13-s + 2.15·14-s − 2.27·16-s − 4.61·17-s − 4.17·19-s + 0.633·20-s + 8.62·22-s − 5.22·23-s − 4.94·25-s − 3.48·26-s + 2.64·28-s − 9.10·29-s − 6.57·31-s − 7.71·32-s − 9.94·34-s + 0.239·35-s − 0.353·37-s − 8.99·38-s + 0.334·40-s + 7.72·41-s + ⋯
L(s)  = 1  + 1.52·2-s + 1.32·4-s + 0.106·5-s + 0.377·7-s + 0.495·8-s + 0.163·10-s + 1.20·11-s − 0.448·13-s + 0.576·14-s − 0.569·16-s − 1.11·17-s − 0.957·19-s + 0.141·20-s + 1.83·22-s − 1.08·23-s − 0.988·25-s − 0.683·26-s + 0.500·28-s − 1.69·29-s − 1.18·31-s − 1.36·32-s − 1.70·34-s + 0.0404·35-s − 0.0581·37-s − 1.45·38-s + 0.0529·40-s + 1.20·41-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 \( 1 - 2.15T + 2T^{2} \)
5 \( 1 - 0.239T + 5T^{2} \)
11 \( 1 - 3.99T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 + 4.61T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 + 5.22T + 23T^{2} \)
29 \( 1 + 9.10T + 29T^{2} \)
31 \( 1 + 6.57T + 31T^{2} \)
37 \( 1 + 0.353T + 37T^{2} \)
41 \( 1 - 7.72T + 41T^{2} \)
43 \( 1 + 7.33T + 43T^{2} \)
47 \( 1 + 3.52T + 47T^{2} \)
53 \( 1 - 0.655T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 7.05T + 61T^{2} \)
67 \( 1 + 0.167T + 67T^{2} \)
71 \( 1 + 6.98T + 71T^{2} \)
73 \( 1 + 2.79T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 6.75T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 1.79T + 97T^{2} \)
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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

−7.22989843521190072365867196363, −6.51261173919209425637670934899, −6.02318747534033094262445361568, −5.34872772235442747434210834735, −4.51178375731342637049001376374, −4.00135965857059444806225029315, −3.50162832179824173452788866040, −2.15306947128086954361354234104, −1.91285762042880458483414421913, 0, 1.91285762042880458483414421913, 2.15306947128086954361354234104, 3.50162832179824173452788866040, 4.00135965857059444806225029315, 4.51178375731342637049001376374, 5.34872772235442747434210834735, 6.02318747534033094262445361568, 6.51261173919209425637670934899, 7.22989843521190072365867196363

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