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  + 2.73·2-s + 5.47·4-s − 4.25·5-s + 7-s + 9.49·8-s − 11.6·10-s − 0.824·11-s − 6.07·13-s + 2.73·14-s + 15.0·16-s + 0.158·17-s + 2.99·19-s − 23.2·20-s − 2.25·22-s + 7.25·23-s + 13.0·25-s − 16.6·26-s + 5.47·28-s + 8.45·29-s + 3.64·31-s + 22.0·32-s + 0.432·34-s − 4.25·35-s − 7.96·37-s + 8.19·38-s − 40.3·40-s + 2.33·41-s + ⋯
L(s)  = 1  + 1.93·2-s + 2.73·4-s − 1.90·5-s + 0.377·7-s + 3.35·8-s − 3.67·10-s − 0.248·11-s − 1.68·13-s + 0.730·14-s + 3.75·16-s + 0.0383·17-s + 0.687·19-s − 5.20·20-s − 0.480·22-s + 1.51·23-s + 2.61·25-s − 3.25·26-s + 1.03·28-s + 1.56·29-s + 0.654·31-s + 3.89·32-s + 0.0741·34-s − 0.719·35-s − 1.30·37-s + 1.32·38-s − 6.38·40-s + 0.364·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.599608561$
$L(\frac12)$  $\approx$  $5.599608561$
$L(\frac{3}{2})$   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 \( 1 - 2.73T + 2T^{2} \)
5 \( 1 + 4.25T + 5T^{2} \)
11 \( 1 + 0.824T + 11T^{2} \)
13 \( 1 + 6.07T + 13T^{2} \)
17 \( 1 - 0.158T + 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 - 7.25T + 23T^{2} \)
29 \( 1 - 8.45T + 29T^{2} \)
31 \( 1 - 3.64T + 31T^{2} \)
37 \( 1 + 7.96T + 37T^{2} \)
41 \( 1 - 2.33T + 41T^{2} \)
43 \( 1 + 7.65T + 43T^{2} \)
47 \( 1 - 1.87T + 47T^{2} \)
53 \( 1 - 4.08T + 53T^{2} \)
59 \( 1 - 5.07T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 8.96T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 3.98T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 2.30T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−7.61314440948378914847965548326, −7.00386457439173431703347222561, −6.56750232813947936203299411243, −5.23895022202927622665735039274, −4.84629680560865756653806070633, −4.53966589559920400597368997798, −3.50400826421841080504352337451, −3.10691142522177840844961790885, −2.31634709768154299594672480993, −0.869172400260465356343915609560, 0.869172400260465356343915609560, 2.31634709768154299594672480993, 3.10691142522177840844961790885, 3.50400826421841080504352337451, 4.53966589559920400597368997798, 4.84629680560865756653806070633, 5.23895022202927622665735039274, 6.56750232813947936203299411243, 7.00386457439173431703347222561, 7.61314440948378914847965548326

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