Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.665·2-s − 1.55·4-s + 3.70·5-s + 7-s + 2.36·8-s − 2.46·10-s − 1.00·11-s + 3.77·13-s − 0.665·14-s + 1.53·16-s − 4.06·17-s − 8.44·19-s − 5.76·20-s + 0.667·22-s − 0.442·23-s + 8.72·25-s − 2.51·26-s − 1.55·28-s − 5.65·29-s + 8.26·31-s − 5.75·32-s + 2.70·34-s + 3.70·35-s − 10.2·37-s + 5.61·38-s + 8.76·40-s + 5.95·41-s + ⋯
L(s)  = 1  − 0.470·2-s − 0.778·4-s + 1.65·5-s + 0.377·7-s + 0.836·8-s − 0.779·10-s − 0.302·11-s + 1.04·13-s − 0.177·14-s + 0.384·16-s − 0.986·17-s − 1.93·19-s − 1.28·20-s + 0.142·22-s − 0.0922·23-s + 1.74·25-s − 0.492·26-s − 0.294·28-s − 1.04·29-s + 1.48·31-s − 1.01·32-s + 0.464·34-s + 0.626·35-s − 1.68·37-s + 0.910·38-s + 1.38·40-s + 0.930·41-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  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.778058750$
$L(\frac12)$  $\approx$  $1.778058750$
$L(\frac{3}{2})$   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 \( 1 + 0.665T + 2T^{2} \)
5 \( 1 - 3.70T + 5T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + 4.06T + 17T^{2} \)
19 \( 1 + 8.44T + 19T^{2} \)
23 \( 1 + 0.442T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 5.95T + 41T^{2} \)
43 \( 1 - 2.00T + 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 0.363T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 3.57T + 61T^{2} \)
67 \( 1 - 3.95T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + 6.24T + 97T^{2} \)
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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

−8.182582882599825908748844750400, −7.08576069632648970325685851405, −6.32643471144037313258455813191, −5.86004482572309278797490398110, −5.02959792963197206946937702840, −4.44340258380957855131926548599, −3.58451846100000807505784501176, −2.22095702839892115641190994232, −1.86609262865125613733798652900, −0.72200862126562241812089895384, 0.72200862126562241812089895384, 1.86609262865125613733798652900, 2.22095702839892115641190994232, 3.58451846100000807505784501176, 4.44340258380957855131926548599, 5.02959792963197206946937702840, 5.86004482572309278797490398110, 6.32643471144037313258455813191, 7.08576069632648970325685851405, 8.182582882599825908748844750400

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