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  + 2.41·2-s + 3.81·4-s + 2.91·5-s − 7-s + 4.37·8-s + 7.02·10-s − 4.84·11-s − 5.92·13-s − 2.41·14-s + 2.91·16-s − 7.70·17-s − 7.16·19-s + 11.1·20-s − 11.6·22-s + 6.74·23-s + 3.48·25-s − 14.2·26-s − 3.81·28-s − 1.16·29-s − 3.19·31-s − 1.71·32-s − 18.5·34-s − 2.91·35-s + 6.23·37-s − 17.2·38-s + 12.7·40-s + 1.69·41-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.90·4-s + 1.30·5-s − 0.377·7-s + 1.54·8-s + 2.22·10-s − 1.45·11-s − 1.64·13-s − 0.644·14-s + 0.728·16-s − 1.86·17-s − 1.64·19-s + 2.48·20-s − 2.48·22-s + 1.40·23-s + 0.697·25-s − 2.80·26-s − 0.720·28-s − 0.216·29-s − 0.573·31-s − 0.303·32-s − 3.18·34-s − 0.492·35-s + 1.02·37-s − 2.80·38-s + 2.01·40-s + 0.264·41-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 \( 1 - 2.41T + 2T^{2} \)
5 \( 1 - 2.91T + 5T^{2} \)
11 \( 1 + 4.84T + 11T^{2} \)
13 \( 1 + 5.92T + 13T^{2} \)
17 \( 1 + 7.70T + 17T^{2} \)
19 \( 1 + 7.16T + 19T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 + 1.16T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 - 6.23T + 37T^{2} \)
41 \( 1 - 1.69T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 4.01T + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 5.30T + 61T^{2} \)
67 \( 1 - 0.727T + 67T^{2} \)
71 \( 1 - 6.03T + 71T^{2} \)
73 \( 1 + 7.68T + 73T^{2} \)
79 \( 1 + 4.20T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 0.0256T + 97T^{2} \)
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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

−7.02187643166151723269281120002, −6.63432142445374020262014022437, −5.89367582681063802909302282055, −5.32474140359363937236744128317, −4.71208172796235369451052890053, −4.22836013672957504117514990807, −2.88677914556933951631716340478, −2.43008946692502491643105431332, −2.06921370312625399927651761357, 0, 2.06921370312625399927651761357, 2.43008946692502491643105431332, 2.88677914556933951631716340478, 4.22836013672957504117514990807, 4.71208172796235369451052890053, 5.32474140359363937236744128317, 5.89367582681063802909302282055, 6.63432142445374020262014022437, 7.02187643166151723269281120002

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