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  + 2.69·2-s + 5.26·4-s − 1.47·5-s + 7-s + 8.80·8-s − 3.97·10-s + 3.56·11-s + 0.225·13-s + 2.69·14-s + 13.2·16-s − 1.78·17-s + 4.11·19-s − 7.76·20-s + 9.60·22-s + 2.13·23-s − 2.82·25-s + 0.609·26-s + 5.26·28-s + 3.51·29-s − 2.58·31-s + 18.0·32-s − 4.81·34-s − 1.47·35-s + 1.05·37-s + 11.0·38-s − 12.9·40-s − 1.46·41-s + ⋯
L(s)  = 1  + 1.90·2-s + 2.63·4-s − 0.659·5-s + 0.377·7-s + 3.11·8-s − 1.25·10-s + 1.07·11-s + 0.0626·13-s + 0.720·14-s + 3.30·16-s − 0.433·17-s + 0.943·19-s − 1.73·20-s + 2.04·22-s + 0.445·23-s − 0.565·25-s + 0.119·26-s + 0.995·28-s + 0.652·29-s − 0.464·31-s + 3.18·32-s − 0.826·34-s − 0.249·35-s + 0.173·37-s + 1.79·38-s − 2.05·40-s − 0.228·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.079917082$
$L(\frac12)$  $\approx$  $8.079917082$
$L(\frac{3}{2})$   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 \( 1 + 1.47T + 5T^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 - 0.225T + 13T^{2} \)
17 \( 1 + 1.78T + 17T^{2} \)
19 \( 1 - 4.11T + 19T^{2} \)
23 \( 1 - 2.13T + 23T^{2} \)
29 \( 1 - 3.51T + 29T^{2} \)
31 \( 1 + 2.58T + 31T^{2} \)
37 \( 1 - 1.05T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 - 7.65T + 43T^{2} \)
47 \( 1 + 6.34T + 47T^{2} \)
53 \( 1 + 0.703T + 53T^{2} \)
59 \( 1 + 4.82T + 59T^{2} \)
61 \( 1 - 7.39T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 4.56T + 79T^{2} \)
83 \( 1 - 3.07T + 83T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.54387640027565099792087969014, −6.89675082731922894882000016873, −6.36009575769071518437780832389, −5.57191084149873352350246804150, −4.92482141648960567743557238942, −4.23287857237950844693371304850, −3.73083763558213004358788522902, −3.03569245503113186440694605647, −2.09134989509520909839454505086, −1.14007739183670673685822199274, 1.14007739183670673685822199274, 2.09134989509520909839454505086, 3.03569245503113186440694605647, 3.73083763558213004358788522902, 4.23287857237950844693371304850, 4.92482141648960567743557238942, 5.57191084149873352350246804150, 6.36009575769071518437780832389, 6.89675082731922894882000016873, 7.54387640027565099792087969014

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