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  − 2.29·2-s + 3.25·4-s + 0.132·5-s + 7-s − 2.88·8-s − 0.304·10-s + 1.38·11-s + 3.00·13-s − 2.29·14-s + 0.0952·16-s − 4.60·17-s − 1.78·19-s + 0.432·20-s − 3.16·22-s − 6.02·23-s − 4.98·25-s − 6.87·26-s + 3.25·28-s − 2.00·29-s + 9.54·31-s + 5.54·32-s + 10.5·34-s + 0.132·35-s + 2.84·37-s + 4.09·38-s − 0.382·40-s − 2.18·41-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.62·4-s + 0.0593·5-s + 0.377·7-s − 1.01·8-s − 0.0962·10-s + 0.416·11-s + 0.832·13-s − 0.612·14-s + 0.0238·16-s − 1.11·17-s − 0.409·19-s + 0.0966·20-s − 0.674·22-s − 1.25·23-s − 0.996·25-s − 1.34·26-s + 0.615·28-s − 0.371·29-s + 1.71·31-s + 0.980·32-s + 1.81·34-s + 0.0224·35-s + 0.467·37-s + 0.663·38-s − 0.0604·40-s − 0.340·41-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 \( 1 + 2.29T + 2T^{2} \)
5 \( 1 - 0.132T + 5T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 + 1.78T + 19T^{2} \)
23 \( 1 + 6.02T + 23T^{2} \)
29 \( 1 + 2.00T + 29T^{2} \)
31 \( 1 - 9.54T + 31T^{2} \)
37 \( 1 - 2.84T + 37T^{2} \)
41 \( 1 + 2.18T + 41T^{2} \)
43 \( 1 - 9.51T + 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 - 8.38T + 53T^{2} \)
59 \( 1 + 7.78T + 59T^{2} \)
61 \( 1 - 3.77T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 - 1.19T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 1.81T + 83T^{2} \)
89 \( 1 - 7.25T + 89T^{2} \)
97 \( 1 - 2.14T + 97T^{2} \)
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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

−7.83815393269048691500439813851, −6.94293383190815849502436797429, −6.33090298287910018678363721017, −5.77907882344571277012594350869, −4.46079105354476977829002301685, −3.99463912841808748066866421825, −2.64763469278711099860502659015, −1.93757126535635117660649444368, −1.13595834172652689410151164670, 0, 1.13595834172652689410151164670, 1.93757126535635117660649444368, 2.64763469278711099860502659015, 3.99463912841808748066866421825, 4.46079105354476977829002301685, 5.77907882344571277012594350869, 6.33090298287910018678363721017, 6.94293383190815849502436797429, 7.83815393269048691500439813851

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