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  − 0.904·2-s − 1.18·4-s − 0.974·5-s − 7-s + 2.87·8-s + 0.881·10-s + 1.96·11-s + 2.74·13-s + 0.904·14-s − 0.238·16-s − 1.83·17-s + 2.02·19-s + 1.15·20-s − 1.77·22-s − 4.95·23-s − 4.04·25-s − 2.47·26-s + 1.18·28-s + 5.22·29-s − 1.18·31-s − 5.53·32-s + 1.66·34-s + 0.974·35-s − 0.458·37-s − 1.83·38-s − 2.80·40-s + 4.63·41-s + ⋯
L(s)  = 1  − 0.639·2-s − 0.591·4-s − 0.435·5-s − 0.377·7-s + 1.01·8-s + 0.278·10-s + 0.592·11-s + 0.760·13-s + 0.241·14-s − 0.0596·16-s − 0.445·17-s + 0.464·19-s + 0.257·20-s − 0.378·22-s − 1.03·23-s − 0.809·25-s − 0.486·26-s + 0.223·28-s + 0.970·29-s − 0.212·31-s − 0.979·32-s + 0.284·34-s + 0.164·35-s − 0.0754·37-s − 0.297·38-s − 0.443·40-s + 0.723·41-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 \( 1 + 0.904T + 2T^{2} \)
5 \( 1 + 0.974T + 5T^{2} \)
11 \( 1 - 1.96T + 11T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 - 2.02T + 19T^{2} \)
23 \( 1 + 4.95T + 23T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + 0.458T + 37T^{2} \)
41 \( 1 - 4.63T + 41T^{2} \)
43 \( 1 + 9.80T + 43T^{2} \)
47 \( 1 + 1.05T + 47T^{2} \)
53 \( 1 - 0.354T + 53T^{2} \)
59 \( 1 - 2.99T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 - 5.79T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.68086239325410809870628272131, −6.89926990654265619693604834135, −6.20482920371951429110836441366, −5.43190056653648181437393730382, −4.48101375066224783849435520801, −3.94772018415713156151341274014, −3.27135651639415773382733644527, −1.99009102828662054218624590582, −1.04294021971578570415275325450, 0, 1.04294021971578570415275325450, 1.99009102828662054218624590582, 3.27135651639415773382733644527, 3.94772018415713156151341274014, 4.48101375066224783849435520801, 5.43190056653648181437393730382, 6.20482920371951429110836441366, 6.89926990654265619693604834135, 7.68086239325410809870628272131

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