Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.910·2-s − 1.17·4-s + 2.06·5-s + 7-s − 2.88·8-s + 1.88·10-s − 4.76·11-s + 2.03·13-s + 0.910·14-s − 0.290·16-s − 0.854·17-s + 2.23·19-s − 2.41·20-s − 4.33·22-s − 5.99·23-s − 0.725·25-s + 1.85·26-s − 1.17·28-s + 5.46·29-s + 3.43·31-s + 5.51·32-s − 0.778·34-s + 2.06·35-s + 1.22·37-s + 2.03·38-s − 5.97·40-s + 7.05·41-s + ⋯
L(s)  = 1  + 0.644·2-s − 0.585·4-s + 0.924·5-s + 0.377·7-s − 1.02·8-s + 0.595·10-s − 1.43·11-s + 0.565·13-s + 0.243·14-s − 0.0725·16-s − 0.207·17-s + 0.513·19-s − 0.540·20-s − 0.924·22-s − 1.25·23-s − 0.145·25-s + 0.364·26-s − 0.221·28-s + 1.01·29-s + 0.616·31-s + 0.974·32-s − 0.133·34-s + 0.349·35-s + 0.201·37-s + 0.330·38-s − 0.944·40-s + 1.10·41-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  =  1
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 0.910T + 2T^{2} \)
5 \( 1 - 2.06T + 5T^{2} \)
11 \( 1 + 4.76T + 11T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 + 0.854T + 17T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 + 5.99T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 - 7.05T + 41T^{2} \)
43 \( 1 + 5.15T + 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 - 1.39T + 53T^{2} \)
59 \( 1 + 8.77T + 59T^{2} \)
61 \( 1 - 6.04T + 61T^{2} \)
67 \( 1 + 5.42T + 67T^{2} \)
71 \( 1 - 1.57T + 71T^{2} \)
73 \( 1 + 9.67T + 73T^{2} \)
79 \( 1 + 4.49T + 79T^{2} \)
83 \( 1 + 8.88T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 1.09T + 97T^{2} \)
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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

−7.61041996852531037361637846242, −6.48013172235609787259214837924, −5.90282278024165624233780655813, −5.40381317182813983453378279994, −4.73442974749308039014387478061, −4.08267028317947110412705812299, −3.05247554831402603684436475666, −2.44600041865091069018085035753, −1.36184883427356093035698038169, 0, 1.36184883427356093035698038169, 2.44600041865091069018085035753, 3.05247554831402603684436475666, 4.08267028317947110412705812299, 4.73442974749308039014387478061, 5.40381317182813983453378279994, 5.90282278024165624233780655813, 6.48013172235609787259214837924, 7.61041996852531037361637846242

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