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  − 1.66·2-s + 0.783·4-s − 3.76·5-s − 7-s + 2.02·8-s + 6.28·10-s + 3.16·11-s − 3.02·13-s + 1.66·14-s − 4.95·16-s − 1.05·17-s + 3.14·19-s − 2.95·20-s − 5.27·22-s + 7.12·23-s + 9.17·25-s + 5.04·26-s − 0.783·28-s − 2.67·29-s − 8.88·31-s + 4.20·32-s + 1.76·34-s + 3.76·35-s + 0.943·37-s − 5.24·38-s − 7.64·40-s − 9.90·41-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.391·4-s − 1.68·5-s − 0.377·7-s + 0.717·8-s + 1.98·10-s + 0.953·11-s − 0.838·13-s + 0.445·14-s − 1.23·16-s − 0.256·17-s + 0.721·19-s − 0.659·20-s − 1.12·22-s + 1.48·23-s + 1.83·25-s + 0.988·26-s − 0.148·28-s − 0.496·29-s − 1.59·31-s + 0.743·32-s + 0.302·34-s + 0.636·35-s + 0.155·37-s − 0.851·38-s − 1.20·40-s − 1.54·41-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 \( 1 + 1.66T + 2T^{2} \)
5 \( 1 + 3.76T + 5T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 + 1.05T + 17T^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + 8.88T + 31T^{2} \)
37 \( 1 - 0.943T + 37T^{2} \)
41 \( 1 + 9.90T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 4.25T + 47T^{2} \)
53 \( 1 - 5.14T + 53T^{2} \)
59 \( 1 - 4.15T + 59T^{2} \)
61 \( 1 - 1.23T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 7.29T + 73T^{2} \)
79 \( 1 - 5.09T + 79T^{2} \)
83 \( 1 - 8.93T + 83T^{2} \)
89 \( 1 + 5.97T + 89T^{2} \)
97 \( 1 + 9.75T + 97T^{2} \)
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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

−7.41053954282002045335359451413, −7.19303162262208333705666010462, −6.60186732222091925743973492239, −5.16540887924653354364908211184, −4.69066491963206788947757412765, −3.69033529662713258187672351156, −3.31120538706073972481402844878, −1.93136039750661812946764356321, −0.847737433780786326464855758232, 0, 0.847737433780786326464855758232, 1.93136039750661812946764356321, 3.31120538706073972481402844878, 3.69033529662713258187672351156, 4.69066491963206788947757412765, 5.16540887924653354364908211184, 6.60186732222091925743973492239, 7.19303162262208333705666010462, 7.41053954282002045335359451413

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