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  − 2.79·2-s + 5.80·4-s + 1.63·5-s + 7-s − 10.6·8-s − 4.57·10-s + 6.47·11-s + 4.75·13-s − 2.79·14-s + 18.0·16-s − 0.645·17-s + 5.60·19-s + 9.49·20-s − 18.0·22-s + 0.0762·23-s − 2.32·25-s − 13.2·26-s + 5.80·28-s − 1.88·29-s + 2.27·31-s − 29.2·32-s + 1.80·34-s + 1.63·35-s − 5.95·37-s − 15.6·38-s − 17.3·40-s + 8.30·41-s + ⋯
L(s)  = 1  − 1.97·2-s + 2.90·4-s + 0.732·5-s + 0.377·7-s − 3.75·8-s − 1.44·10-s + 1.95·11-s + 1.31·13-s − 0.746·14-s + 4.51·16-s − 0.156·17-s + 1.28·19-s + 2.12·20-s − 3.85·22-s + 0.0158·23-s − 0.464·25-s − 2.60·26-s + 1.09·28-s − 0.349·29-s + 0.408·31-s − 5.16·32-s + 0.309·34-s + 0.276·35-s − 0.978·37-s − 2.53·38-s − 2.74·40-s + 1.29·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.450594751$
$L(\frac12)$  $\approx$  $1.450594751$
$L(\frac{3}{2})$   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 \( 1 + 2.79T + 2T^{2} \)
5 \( 1 - 1.63T + 5T^{2} \)
11 \( 1 - 6.47T + 11T^{2} \)
13 \( 1 - 4.75T + 13T^{2} \)
17 \( 1 + 0.645T + 17T^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 - 0.0762T + 23T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 - 2.27T + 31T^{2} \)
37 \( 1 + 5.95T + 37T^{2} \)
41 \( 1 - 8.30T + 41T^{2} \)
43 \( 1 - 0.409T + 43T^{2} \)
47 \( 1 + 5.35T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 8.26T + 59T^{2} \)
61 \( 1 + 4.31T + 61T^{2} \)
67 \( 1 - 7.25T + 67T^{2} \)
71 \( 1 + 3.65T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 4.61T + 89T^{2} \)
97 \( 1 + 8.08T + 97T^{2} \)
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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

−8.086754384484608634745436542105, −7.17027115407796497467325034582, −6.74105290192718151895193434847, −6.01509137351068113922208965640, −5.57107754449854422523548683956, −3.98565297495991608840013700974, −3.25385065471554827099906507345, −2.13834391695643256382448506840, −1.41699001079377265647565537845, −0.930433836898779631668769715138, 0.930433836898779631668769715138, 1.41699001079377265647565537845, 2.13834391695643256382448506840, 3.25385065471554827099906507345, 3.98565297495991608840013700974, 5.57107754449854422523548683956, 6.01509137351068113922208965640, 6.74105290192718151895193434847, 7.17027115407796497467325034582, 8.086754384484608634745436542105

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