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  − 1.20·2-s − 0.540·4-s − 0.486·5-s + 7-s + 3.06·8-s + 0.588·10-s − 3.33·11-s + 2.96·13-s − 1.20·14-s − 2.62·16-s − 0.691·17-s + 0.556·19-s + 0.262·20-s + 4.03·22-s − 4.97·23-s − 4.76·25-s − 3.57·26-s − 0.540·28-s − 1.54·29-s − 1.24·31-s − 2.96·32-s + 0.836·34-s − 0.486·35-s − 7.43·37-s − 0.672·38-s − 1.49·40-s + 5.74·41-s + ⋯
L(s)  = 1  − 0.854·2-s − 0.270·4-s − 0.217·5-s + 0.377·7-s + 1.08·8-s + 0.186·10-s − 1.00·11-s + 0.821·13-s − 0.322·14-s − 0.656·16-s − 0.167·17-s + 0.127·19-s + 0.0588·20-s + 0.859·22-s − 1.03·23-s − 0.952·25-s − 0.701·26-s − 0.102·28-s − 0.286·29-s − 0.222·31-s − 0.523·32-s + 0.143·34-s − 0.0822·35-s − 1.22·37-s − 0.109·38-s − 0.236·40-s + 0.897·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7363129649$
$L(\frac12)$  $\approx$  $0.7363129649$
$L(\frac{3}{2})$   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 \( 1 + 1.20T + 2T^{2} \)
5 \( 1 + 0.486T + 5T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 + 0.691T + 17T^{2} \)
19 \( 1 - 0.556T + 19T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + 1.54T + 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 - 9.66T + 43T^{2} \)
47 \( 1 + 4.42T + 47T^{2} \)
53 \( 1 - 3.25T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 9.14T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 0.539T + 83T^{2} \)
89 \( 1 - 9.15T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−7.906670752013784400446987423482, −7.55060415978254284842924152654, −6.58579227616716074775782090110, −5.67176250893525302861214842420, −5.12075274952762021180955676205, −4.17973964722401990446684329223, −3.67757054886173881833238968189, −2.41189317064705981281211216037, −1.61031829883420428138172600582, −0.49603267558632046328035526728, 0.49603267558632046328035526728, 1.61031829883420428138172600582, 2.41189317064705981281211216037, 3.67757054886173881833238968189, 4.17973964722401990446684329223, 5.12075274952762021180955676205, 5.67176250893525302861214842420, 6.58579227616716074775782090110, 7.55060415978254284842924152654, 7.906670752013784400446987423482

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