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  − 1.70·2-s + 0.914·4-s − 1.08·5-s + 7-s + 1.85·8-s + 1.85·10-s − 1.58·11-s − 5.55·13-s − 1.70·14-s − 4.99·16-s + 3.38·17-s − 4.84·19-s − 0.994·20-s + 2.69·22-s − 7.35·23-s − 3.81·25-s + 9.48·26-s + 0.914·28-s − 4.01·29-s − 8.96·31-s + 4.81·32-s − 5.77·34-s − 1.08·35-s − 3.94·37-s + 8.26·38-s − 2.01·40-s − 6.40·41-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.457·4-s − 0.486·5-s + 0.377·7-s + 0.655·8-s + 0.586·10-s − 0.476·11-s − 1.54·13-s − 0.456·14-s − 1.24·16-s + 0.820·17-s − 1.11·19-s − 0.222·20-s + 0.575·22-s − 1.53·23-s − 0.763·25-s + 1.86·26-s + 0.172·28-s − 0.745·29-s − 1.60·31-s + 0.851·32-s − 0.990·34-s − 0.183·35-s − 0.647·37-s + 1.34·38-s − 0.318·40-s − 1.00·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1506037682$
$L(\frac12)$  $\approx$  $0.1506037682$
$L(\frac{3}{2})$   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 \( 1 + 1.70T + 2T^{2} \)
5 \( 1 + 1.08T + 5T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 - 3.38T + 17T^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 + 7.35T + 23T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + 6.40T + 41T^{2} \)
43 \( 1 + 8.32T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2.86T + 53T^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 5.60T + 79T^{2} \)
83 \( 1 - 6.19T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 4.99T + 97T^{2} \)
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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

−7.82203954375560564719379416262, −7.49603506410659620294232346461, −6.84756334904962785366876147260, −5.71436500111919490550401951644, −5.06951296571166587250989574654, −4.26885228052738799832666176843, −3.54457133743801312635079940677, −2.20663277995600284180304470127, −1.77628071396318123498279532906, −0.22216555930038465445484228630, 0.22216555930038465445484228630, 1.77628071396318123498279532906, 2.20663277995600284180304470127, 3.54457133743801312635079940677, 4.26885228052738799832666176843, 5.06951296571166587250989574654, 5.71436500111919490550401951644, 6.84756334904962785366876147260, 7.49603506410659620294232346461, 7.82203954375560564719379416262

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