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  + 1.24·2-s − 0.455·4-s + 3.33·5-s + 7-s − 3.05·8-s + 4.14·10-s − 2.90·11-s − 4.59·13-s + 1.24·14-s − 2.88·16-s + 5.29·17-s + 1.86·19-s − 1.51·20-s − 3.61·22-s − 5.03·23-s + 6.11·25-s − 5.70·26-s − 0.455·28-s − 3.69·29-s + 0.968·31-s + 2.52·32-s + 6.57·34-s + 3.33·35-s − 10.5·37-s + 2.32·38-s − 10.1·40-s − 11.5·41-s + ⋯
L(s)  = 1  + 0.878·2-s − 0.227·4-s + 1.49·5-s + 0.377·7-s − 1.07·8-s + 1.31·10-s − 0.876·11-s − 1.27·13-s + 0.332·14-s − 0.720·16-s + 1.28·17-s + 0.428·19-s − 0.339·20-s − 0.770·22-s − 1.04·23-s + 1.22·25-s − 1.11·26-s − 0.0860·28-s − 0.686·29-s + 0.173·31-s + 0.445·32-s + 1.12·34-s + 0.563·35-s − 1.73·37-s + 0.376·38-s − 1.60·40-s − 1.80·41-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 \( 1 - 1.24T + 2T^{2} \)
5 \( 1 - 3.33T + 5T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 + 4.59T + 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
23 \( 1 + 5.03T + 23T^{2} \)
29 \( 1 + 3.69T + 29T^{2} \)
31 \( 1 - 0.968T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 0.666T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + 9.88T + 71T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 0.493T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.38122900819224753038185568029, −6.62238150690612575672200777024, −5.67701691357897398489990150570, −5.34675730370273679577551964375, −5.03032160131648042704047405169, −3.96713110418069451655333286871, −3.08120065647249204041561772042, −2.38091953429631983289000196751, −1.55690183422270188191248524935, 0, 1.55690183422270188191248524935, 2.38091953429631983289000196751, 3.08120065647249204041561772042, 3.96713110418069451655333286871, 5.03032160131648042704047405169, 5.34675730370273679577551964375, 5.67701691357897398489990150570, 6.62238150690612575672200777024, 7.38122900819224753038185568029

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