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  + 2.23·2-s + 3.01·4-s + 0.715·5-s − 7-s + 2.26·8-s + 1.60·10-s + 3.48·11-s − 3.88·13-s − 2.23·14-s − 0.957·16-s − 2.02·17-s + 1.32·19-s + 2.15·20-s + 7.80·22-s − 4.77·23-s − 4.48·25-s − 8.68·26-s − 3.01·28-s − 8.26·29-s − 4.73·31-s − 6.66·32-s − 4.52·34-s − 0.715·35-s − 10.4·37-s + 2.96·38-s + 1.61·40-s + 0.827·41-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s + 0.319·5-s − 0.377·7-s + 0.799·8-s + 0.506·10-s + 1.05·11-s − 1.07·13-s − 0.598·14-s − 0.239·16-s − 0.490·17-s + 0.304·19-s + 0.481·20-s + 1.66·22-s − 0.995·23-s − 0.897·25-s − 1.70·26-s − 0.568·28-s − 1.53·29-s − 0.849·31-s − 1.17·32-s − 0.775·34-s − 0.120·35-s − 1.72·37-s + 0.481·38-s + 0.255·40-s + 0.129·41-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 \( 1 - 2.23T + 2T^{2} \)
5 \( 1 - 0.715T + 5T^{2} \)
11 \( 1 - 3.48T + 11T^{2} \)
13 \( 1 + 3.88T + 13T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + 4.77T + 23T^{2} \)
29 \( 1 + 8.26T + 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 0.827T + 41T^{2} \)
43 \( 1 - 2.69T + 43T^{2} \)
47 \( 1 + 9.44T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + 6.64T + 59T^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 8.80T + 71T^{2} \)
73 \( 1 - 6.52T + 73T^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 - 6.83T + 97T^{2} \)
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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

−7.03418820347254859516998396037, −6.74263256053680530159600382952, −5.83634133300945797410847080607, −5.44356645431918776489702981774, −4.65317298833683840214980405839, −3.78926349184781116773424050445, −3.52935906831146021486818010839, −2.31204624934484181631983322806, −1.83285578304919313313607525603, 0, 1.83285578304919313313607525603, 2.31204624934484181631983322806, 3.52935906831146021486818010839, 3.78926349184781116773424050445, 4.65317298833683840214980405839, 5.44356645431918776489702981774, 5.83634133300945797410847080607, 6.74263256053680530159600382952, 7.03418820347254859516998396037

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