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  − 1.18·2-s − 0.607·4-s + 4.43·5-s − 7-s + 3.07·8-s − 5.23·10-s − 5.41·11-s − 2.38·13-s + 1.18·14-s − 2.41·16-s + 2.36·17-s − 2.75·19-s − 2.69·20-s + 6.39·22-s − 5.73·23-s + 14.6·25-s + 2.81·26-s + 0.607·28-s − 3.90·29-s + 2.83·31-s − 3.30·32-s − 2.78·34-s − 4.43·35-s − 3.58·37-s + 3.25·38-s + 13.6·40-s − 0.707·41-s + ⋯
L(s)  = 1  − 0.834·2-s − 0.303·4-s + 1.98·5-s − 0.377·7-s + 1.08·8-s − 1.65·10-s − 1.63·11-s − 0.661·13-s + 0.315·14-s − 0.604·16-s + 0.572·17-s − 0.632·19-s − 0.602·20-s + 1.36·22-s − 1.19·23-s + 2.93·25-s + 0.551·26-s + 0.114·28-s − 0.725·29-s + 0.509·31-s − 0.583·32-s − 0.477·34-s − 0.750·35-s − 0.589·37-s + 0.527·38-s + 2.15·40-s − 0.110·41-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 \( 1 + 1.18T + 2T^{2} \)
5 \( 1 - 4.43T + 5T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + 2.38T + 13T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 + 2.75T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 + 3.58T + 37T^{2} \)
41 \( 1 + 0.707T + 41T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 15.4T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 - 5.77T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−7.50582328821994848720363834019, −6.98326093595861854495764693752, −5.90124016050090547710334333033, −5.54710185984756811595306201402, −4.93548756603028870646867939392, −3.94837432797236099325285836928, −2.51323680077382289971597742197, −2.32440059651922780510710434731, −1.20377590485720792860619938233, 0, 1.20377590485720792860619938233, 2.32440059651922780510710434731, 2.51323680077382289971597742197, 3.94837432797236099325285836928, 4.93548756603028870646867939392, 5.54710185984756811595306201402, 5.90124016050090547710334333033, 6.98326093595861854495764693752, 7.50582328821994848720363834019

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