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  − 0.823·2-s − 1.32·4-s + 2.87·5-s + 7-s + 2.73·8-s − 2.37·10-s + 2.49·11-s − 3.14·13-s − 0.823·14-s + 0.390·16-s + 2.04·17-s + 3.80·19-s − 3.80·20-s − 2.05·22-s + 0.178·23-s + 3.28·25-s + 2.58·26-s − 1.32·28-s − 7.77·29-s + 8.24·31-s − 5.79·32-s − 1.68·34-s + 2.87·35-s + 0.0498·37-s − 3.12·38-s + 7.87·40-s + 2.14·41-s + ⋯
L(s)  = 1  − 0.582·2-s − 0.660·4-s + 1.28·5-s + 0.377·7-s + 0.967·8-s − 0.749·10-s + 0.752·11-s − 0.871·13-s − 0.220·14-s + 0.0976·16-s + 0.495·17-s + 0.871·19-s − 0.850·20-s − 0.438·22-s + 0.0372·23-s + 0.656·25-s + 0.507·26-s − 0.249·28-s − 1.44·29-s + 1.48·31-s − 1.02·32-s − 0.288·34-s + 0.486·35-s + 0.00819·37-s − 0.507·38-s + 1.24·40-s + 0.335·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.880205767$
$L(\frac12)$  $\approx$  $1.880205767$
$L(\frac{3}{2})$   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 \( 1 + 0.823T + 2T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
11 \( 1 - 2.49T + 11T^{2} \)
13 \( 1 + 3.14T + 13T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
19 \( 1 - 3.80T + 19T^{2} \)
23 \( 1 - 0.178T + 23T^{2} \)
29 \( 1 + 7.77T + 29T^{2} \)
31 \( 1 - 8.24T + 31T^{2} \)
37 \( 1 - 0.0498T + 37T^{2} \)
41 \( 1 - 2.14T + 41T^{2} \)
43 \( 1 - 4.51T + 43T^{2} \)
47 \( 1 - 0.391T + 47T^{2} \)
53 \( 1 - 8.31T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 8.25T + 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 + 8.03T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + 5.72T + 79T^{2} \)
83 \( 1 - 8.13T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 1.62T + 97T^{2} \)
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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

−7.75078721973351101499165370205, −7.41612738547200905693656005110, −6.41380192690865010432339262149, −5.69259671517140951930693101610, −5.11113966585809370074584057347, −4.43068335933744138393176894823, −3.51600469640004131327516683857, −2.41846283976401927739834411990, −1.60329008823764246808058810278, −0.803309115819676665775444441180, 0.803309115819676665775444441180, 1.60329008823764246808058810278, 2.41846283976401927739834411990, 3.51600469640004131327516683857, 4.43068335933744138393176894823, 5.11113966585809370074584057347, 5.69259671517140951930693101610, 6.41380192690865010432339262149, 7.41612738547200905693656005110, 7.75078721973351101499165370205

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