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  + 1.63·2-s + 0.672·4-s − 0.107·5-s − 7-s − 2.16·8-s − 0.176·10-s + 1.23·11-s − 2.67·13-s − 1.63·14-s − 4.89·16-s + 6.14·17-s + 1.57·19-s − 0.0725·20-s + 2.01·22-s + 5.97·23-s − 4.98·25-s − 4.37·26-s − 0.672·28-s − 9.60·29-s + 1.81·31-s − 3.65·32-s + 10.0·34-s + 0.107·35-s + 7.63·37-s + 2.57·38-s + 0.233·40-s − 4.73·41-s + ⋯
L(s)  = 1  + 1.15·2-s + 0.336·4-s − 0.0482·5-s − 0.377·7-s − 0.767·8-s − 0.0557·10-s + 0.372·11-s − 0.742·13-s − 0.436·14-s − 1.22·16-s + 1.48·17-s + 0.361·19-s − 0.0162·20-s + 0.430·22-s + 1.24·23-s − 0.997·25-s − 0.858·26-s − 0.127·28-s − 1.78·29-s + 0.325·31-s − 0.646·32-s + 1.72·34-s + 0.0182·35-s + 1.25·37-s + 0.418·38-s + 0.0369·40-s − 0.739·41-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 \( 1 - 1.63T + 2T^{2} \)
5 \( 1 + 0.107T + 5T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 - 6.14T + 17T^{2} \)
19 \( 1 - 1.57T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 + 9.60T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
37 \( 1 - 7.63T + 37T^{2} \)
41 \( 1 + 4.73T + 41T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 - 5.68T + 59T^{2} \)
61 \( 1 + 8.36T + 61T^{2} \)
67 \( 1 + 4.40T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 8.13T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.52494426370192881932748569904, −6.53654113888640337853343536282, −5.96302325630134959325509906427, −5.30065373280014348779949009966, −4.73784561958481588264457632247, −3.85652552346792924088143307873, −3.29688941284644146068349327620, −2.62397812562791776893873251529, −1.38714402249921382784496680960, 0, 1.38714402249921382784496680960, 2.62397812562791776893873251529, 3.29688941284644146068349327620, 3.85652552346792924088143307873, 4.73784561958481588264457632247, 5.30065373280014348779949009966, 5.96302325630134959325509906427, 6.53654113888640337853343536282, 7.52494426370192881932748569904

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