Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52·2-s + 0.325·4-s + 1.71·5-s + 7-s − 2.55·8-s + 2.61·10-s + 1.69·11-s − 1.16·13-s + 1.52·14-s − 4.54·16-s − 5.50·17-s − 0.828·19-s + 0.557·20-s + 2.58·22-s + 6.42·23-s − 2.06·25-s − 1.78·26-s + 0.325·28-s + 2.43·29-s − 0.583·31-s − 1.82·32-s − 8.39·34-s + 1.71·35-s + 11.0·37-s − 1.26·38-s − 4.37·40-s + 0.0442·41-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.162·4-s + 0.765·5-s + 0.377·7-s − 0.902·8-s + 0.825·10-s + 0.510·11-s − 0.323·13-s + 0.407·14-s − 1.13·16-s − 1.33·17-s − 0.190·19-s + 0.124·20-s + 0.550·22-s + 1.33·23-s − 0.413·25-s − 0.349·26-s + 0.0615·28-s + 0.452·29-s − 0.104·31-s − 0.322·32-s − 1.43·34-s + 0.289·35-s + 1.81·37-s − 0.204·38-s − 0.691·40-s + 0.00690·41-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  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.788317334$
$L(\frac12)$  $\approx$  $3.788317334$
$L(\frac{3}{2})$   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 \( 1 - 1.52T + 2T^{2} \)
5 \( 1 - 1.71T + 5T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 + 1.16T + 13T^{2} \)
17 \( 1 + 5.50T + 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 - 6.42T + 23T^{2} \)
29 \( 1 - 2.43T + 29T^{2} \)
31 \( 1 + 0.583T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 0.0442T + 41T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + 3.58T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + 1.61T + 59T^{2} \)
61 \( 1 + 5.74T + 61T^{2} \)
67 \( 1 + 0.393T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 7.94T + 73T^{2} \)
79 \( 1 - 3.92T + 79T^{2} \)
83 \( 1 + 6.36T + 83T^{2} \)
89 \( 1 - 5.77T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−7.65746932656802218109257213004, −6.92959859968198986319974281826, −6.13211829673005614417378500443, −5.83154319308645570086619791417, −4.78442936482389259079107455100, −4.53850671022921338487339313339, −3.68075722340534874310805226252, −2.67391388820953156575868994943, −2.13090315174417702118542184510, −0.816509380948081526736995027816, 0.816509380948081526736995027816, 2.13090315174417702118542184510, 2.67391388820953156575868994943, 3.68075722340534874310805226252, 4.53850671022921338487339313339, 4.78442936482389259079107455100, 5.83154319308645570086619791417, 6.13211829673005614417378500443, 6.92959859968198986319974281826, 7.65746932656802218109257213004

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