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  + 0.242·2-s − 1.94·4-s − 2.79·5-s − 7-s − 0.955·8-s − 0.678·10-s − 4.59·11-s − 4.92·13-s − 0.242·14-s + 3.65·16-s + 2.21·17-s + 0.299·19-s + 5.42·20-s − 1.11·22-s + 2.16·23-s + 2.82·25-s − 1.19·26-s + 1.94·28-s + 5.57·29-s + 6.00·31-s + 2.79·32-s + 0.536·34-s + 2.79·35-s + 8.81·37-s + 0.0726·38-s + 2.67·40-s + 10.5·41-s + ⋯
L(s)  = 1  + 0.171·2-s − 0.970·4-s − 1.25·5-s − 0.377·7-s − 0.337·8-s − 0.214·10-s − 1.38·11-s − 1.36·13-s − 0.0648·14-s + 0.912·16-s + 0.536·17-s + 0.0687·19-s + 1.21·20-s − 0.237·22-s + 0.452·23-s + 0.564·25-s − 0.234·26-s + 0.366·28-s + 1.03·29-s + 1.07·31-s + 0.494·32-s + 0.0920·34-s + 0.472·35-s + 1.44·37-s + 0.0117·38-s + 0.422·40-s + 1.64·41-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 \( 1 - 0.242T + 2T^{2} \)
5 \( 1 + 2.79T + 5T^{2} \)
11 \( 1 + 4.59T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 - 0.299T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 - 6.00T + 31T^{2} \)
37 \( 1 - 8.81T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 6.69T + 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 - 1.04T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 0.161T + 61T^{2} \)
67 \( 1 - 2.60T + 67T^{2} \)
71 \( 1 - 2.49T + 71T^{2} \)
73 \( 1 + 5.94T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 + 9.33T + 89T^{2} \)
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

−7.71175070464681845874775289282, −6.97614053509692112311705052865, −5.97203712262921038394931826970, −5.14368342514023857663611258238, −4.66409853821561191826423502874, −4.05983001990012939739226614589, −3.05847089138549230398302857214, −2.65231110362184489735320613962, −0.824712640035015421049780521359, 0, 0.824712640035015421049780521359, 2.65231110362184489735320613962, 3.05847089138549230398302857214, 4.05983001990012939739226614589, 4.66409853821561191826423502874, 5.14368342514023857663611258238, 5.97203712262921038394931826970, 6.97614053509692112311705052865, 7.71175070464681845874775289282

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