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  − 2.69·2-s + 5.26·4-s + 3.35·5-s − 7-s − 8.81·8-s − 9.05·10-s − 2.89·11-s − 2.10·13-s + 2.69·14-s + 13.2·16-s + 6.29·17-s − 0.167·19-s + 17.6·20-s + 7.80·22-s + 4.03·23-s + 6.28·25-s + 5.68·26-s − 5.26·28-s − 0.392·29-s − 7.54·31-s − 18.0·32-s − 16.9·34-s − 3.35·35-s − 2.42·37-s + 0.450·38-s − 29.6·40-s + 0.160·41-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.63·4-s + 1.50·5-s − 0.377·7-s − 3.11·8-s − 2.86·10-s − 0.872·11-s − 0.584·13-s + 0.720·14-s + 3.30·16-s + 1.52·17-s − 0.0383·19-s + 3.95·20-s + 1.66·22-s + 0.841·23-s + 1.25·25-s + 1.11·26-s − 0.995·28-s − 0.0727·29-s − 1.35·31-s − 3.18·32-s − 2.90·34-s − 0.567·35-s − 0.399·37-s + 0.0731·38-s − 4.68·40-s + 0.0251·41-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 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 - 3.35T + 5T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 + 2.10T + 13T^{2} \)
17 \( 1 - 6.29T + 17T^{2} \)
19 \( 1 + 0.167T + 19T^{2} \)
23 \( 1 - 4.03T + 23T^{2} \)
29 \( 1 + 0.392T + 29T^{2} \)
31 \( 1 + 7.54T + 31T^{2} \)
37 \( 1 + 2.42T + 37T^{2} \)
41 \( 1 - 0.160T + 41T^{2} \)
43 \( 1 + 6.15T + 43T^{2} \)
47 \( 1 + 8.81T + 47T^{2} \)
53 \( 1 - 2.87T + 53T^{2} \)
59 \( 1 - 3.45T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 9.36T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 3.36T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−7.56734841155529496080696267410, −7.03395660043013078314971111989, −6.34450089249562463294815539038, −5.58321091302913426031762943851, −5.17254428232854490536115960669, −3.32955328759689101793908242379, −2.71646524715798009941569587156, −1.92498085767698358500431788361, −1.23109092902642923581321582119, 0, 1.23109092902642923581321582119, 1.92498085767698358500431788361, 2.71646524715798009941569587156, 3.32955328759689101793908242379, 5.17254428232854490536115960669, 5.58321091302913426031762943851, 6.34450089249562463294815539038, 7.03395660043013078314971111989, 7.56734841155529496080696267410

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