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  + 0.407·2-s − 1.83·4-s + 1.08·5-s + 7-s − 1.56·8-s + 0.444·10-s − 2.30·11-s − 1.28·13-s + 0.407·14-s + 3.02·16-s − 3.85·17-s − 5.48·19-s − 1.99·20-s − 0.938·22-s − 5.15·23-s − 3.81·25-s − 0.523·26-s − 1.83·28-s + 3.35·29-s − 0.261·31-s + 4.36·32-s − 1.57·34-s + 1.08·35-s + 4.76·37-s − 2.23·38-s − 1.70·40-s − 4.53·41-s + ⋯
L(s)  = 1  + 0.288·2-s − 0.916·4-s + 0.487·5-s + 0.377·7-s − 0.552·8-s + 0.140·10-s − 0.693·11-s − 0.355·13-s + 0.109·14-s + 0.757·16-s − 0.936·17-s − 1.25·19-s − 0.446·20-s − 0.200·22-s − 1.07·23-s − 0.762·25-s − 0.102·26-s − 0.346·28-s + 0.622·29-s − 0.0470·31-s + 0.771·32-s − 0.270·34-s + 0.184·35-s + 0.783·37-s − 0.362·38-s − 0.269·40-s − 0.708·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.253315246$
$L(\frac12)$  $\approx$  $1.253315246$
$L(\frac{3}{2})$   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 \( 1 - 0.407T + 2T^{2} \)
5 \( 1 - 1.08T + 5T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 + 3.85T + 17T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 + 5.15T + 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 + 0.261T + 31T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 + 0.491T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 1.96T + 59T^{2} \)
61 \( 1 - 0.0291T + 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 + 1.95T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 - 9.68T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 2.44T + 97T^{2} \)
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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

−8.188880660107996932404996862384, −7.10475256114005153255523492033, −6.25805521210753868924064134412, −5.70363634934105569543793381437, −4.95973888949513114572995586438, −4.35876713787497010301074816839, −3.74889404333134458962982874095, −2.54513614678883617539028117699, −1.97580459245144797338721717463, −0.51215162268467314258008314380, 0.51215162268467314258008314380, 1.97580459245144797338721717463, 2.54513614678883617539028117699, 3.74889404333134458962982874095, 4.35876713787497010301074816839, 4.95973888949513114572995586438, 5.70363634934105569543793381437, 6.25805521210753868924064134412, 7.10475256114005153255523492033, 8.188880660107996932404996862384

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