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  − 0.203·2-s − 1.95·4-s − 2.16·5-s − 7-s + 0.804·8-s + 0.440·10-s + 2.03·11-s + 1.83·13-s + 0.203·14-s + 3.75·16-s − 2.26·17-s − 5.79·19-s + 4.24·20-s − 0.413·22-s + 2.00·23-s − 0.298·25-s − 0.373·26-s + 1.95·28-s − 2.70·29-s + 0.925·31-s − 2.37·32-s + 0.460·34-s + 2.16·35-s + 7.11·37-s + 1.17·38-s − 1.74·40-s − 3.83·41-s + ⋯
L(s)  = 1  − 0.143·2-s − 0.979·4-s − 0.969·5-s − 0.377·7-s + 0.284·8-s + 0.139·10-s + 0.613·11-s + 0.509·13-s + 0.0543·14-s + 0.938·16-s − 0.549·17-s − 1.32·19-s + 0.949·20-s − 0.0882·22-s + 0.417·23-s − 0.0596·25-s − 0.0732·26-s + 0.370·28-s − 0.501·29-s + 0.166·31-s − 0.419·32-s + 0.0789·34-s + 0.366·35-s + 1.16·37-s + 0.191·38-s − 0.275·40-s − 0.598·41-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 \( 1 + 0.203T + 2T^{2} \)
5 \( 1 + 2.16T + 5T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 - 1.83T + 13T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + 2.70T + 29T^{2} \)
31 \( 1 - 0.925T + 31T^{2} \)
37 \( 1 - 7.11T + 37T^{2} \)
41 \( 1 + 3.83T + 41T^{2} \)
43 \( 1 + 0.434T + 43T^{2} \)
47 \( 1 - 4.07T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + 9.25T + 61T^{2} \)
67 \( 1 - 6.88T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 4.78T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 4.97T + 83T^{2} \)
89 \( 1 + 1.11T + 89T^{2} \)
97 \( 1 - 7.04T + 97T^{2} \)
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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

−7.69229956251577418259678015876, −6.75329728418172404372917455618, −6.23689297832912568329133077853, −5.28323305142307620813988857560, −4.45181147303834225607573211679, −3.94829717506003945262634851725, −3.43510736094344366292944406850, −2.20669523188218073146410777631, −0.949460666662730038074772411714, 0, 0.949460666662730038074772411714, 2.20669523188218073146410777631, 3.43510736094344366292944406850, 3.94829717506003945262634851725, 4.45181147303834225607573211679, 5.28323305142307620813988857560, 6.23689297832912568329133077853, 6.75329728418172404372917455618, 7.69229956251577418259678015876

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