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  + 2.77·2-s + 5.67·4-s + 3.69·5-s + 7-s + 10.1·8-s + 10.2·10-s + 5.84·11-s − 6.83·13-s + 2.77·14-s + 16.8·16-s − 2.91·17-s − 0.728·19-s + 20.9·20-s + 16.1·22-s − 8.42·23-s + 8.64·25-s − 18.9·26-s + 5.67·28-s − 5.16·29-s − 0.147·31-s + 26.3·32-s − 8.06·34-s + 3.69·35-s + 0.220·37-s − 2.01·38-s + 37.6·40-s − 6.63·41-s + ⋯
L(s)  = 1  + 1.95·2-s + 2.83·4-s + 1.65·5-s + 0.377·7-s + 3.59·8-s + 3.23·10-s + 1.76·11-s − 1.89·13-s + 0.740·14-s + 4.21·16-s − 0.706·17-s − 0.167·19-s + 4.68·20-s + 3.45·22-s − 1.75·23-s + 1.72·25-s − 3.71·26-s + 1.07·28-s − 0.959·29-s − 0.0264·31-s + 4.65·32-s − 1.38·34-s + 0.624·35-s + 0.0362·37-s − 0.327·38-s + 5.94·40-s − 1.03·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $11.75134421$
$L(\frac12)$  $\approx$  $11.75134421$
$L(\frac{3}{2})$   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 \( 1 - 2.77T + 2T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 - 5.84T + 11T^{2} \)
13 \( 1 + 6.83T + 13T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
19 \( 1 + 0.728T + 19T^{2} \)
23 \( 1 + 8.42T + 23T^{2} \)
29 \( 1 + 5.16T + 29T^{2} \)
31 \( 1 + 0.147T + 31T^{2} \)
37 \( 1 - 0.220T + 37T^{2} \)
41 \( 1 + 6.63T + 41T^{2} \)
43 \( 1 - 2.26T + 43T^{2} \)
47 \( 1 - 2.62T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 5.14T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + 6.18T + 73T^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
83 \( 1 + 5.02T + 83T^{2} \)
89 \( 1 + 9.91T + 89T^{2} \)
97 \( 1 + 3.08T + 97T^{2} \)
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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

−7.24585189796299149639883431480, −6.88261728759624204035143197053, −6.20279356944685060344600121514, −5.65700972384137776373789896866, −5.09516548322317482437352165215, −4.31551889429669618745637729573, −3.80471857078877337268110718171, −2.56324605659848324899047390290, −2.11917164223080557863305919519, −1.52377047116193602027424215030, 1.52377047116193602027424215030, 2.11917164223080557863305919519, 2.56324605659848324899047390290, 3.80471857078877337268110718171, 4.31551889429669618745637729573, 5.09516548322317482437352165215, 5.65700972384137776373789896866, 6.20279356944685060344600121514, 6.88261728759624204035143197053, 7.24585189796299149639883431480

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