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  + 1.30·2-s − 0.309·4-s − 3.89·5-s + 7-s − 3.00·8-s − 5.06·10-s + 2.39·11-s + 6.75·13-s + 1.30·14-s − 3.28·16-s − 2.21·17-s + 2.24·19-s + 1.20·20-s + 3.11·22-s − 6.95·23-s + 10.1·25-s + 8.77·26-s − 0.309·28-s + 4.13·29-s − 7.14·31-s + 1.73·32-s − 2.88·34-s − 3.89·35-s − 5.98·37-s + 2.91·38-s + 11.6·40-s − 4.17·41-s + ⋯
L(s)  = 1  + 0.919·2-s − 0.154·4-s − 1.74·5-s + 0.377·7-s − 1.06·8-s − 1.60·10-s + 0.722·11-s + 1.87·13-s + 0.347·14-s − 0.821·16-s − 0.537·17-s + 0.514·19-s + 0.269·20-s + 0.664·22-s − 1.45·23-s + 2.03·25-s + 1.72·26-s − 0.0584·28-s + 0.768·29-s − 1.28·31-s + 0.306·32-s − 0.494·34-s − 0.658·35-s − 0.984·37-s + 0.473·38-s + 1.84·40-s − 0.652·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.710675007$
$L(\frac12)$  $\approx$  $1.710675007$
$L(\frac{3}{2})$   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 \( 1 - 1.30T + 2T^{2} \)
5 \( 1 + 3.89T + 5T^{2} \)
11 \( 1 - 2.39T + 11T^{2} \)
13 \( 1 - 6.75T + 13T^{2} \)
17 \( 1 + 2.21T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 + 7.14T + 31T^{2} \)
37 \( 1 + 5.98T + 37T^{2} \)
41 \( 1 + 4.17T + 41T^{2} \)
43 \( 1 + 6.77T + 43T^{2} \)
47 \( 1 + 8.85T + 47T^{2} \)
53 \( 1 - 1.24T + 53T^{2} \)
59 \( 1 + 0.805T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 0.850T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 3.01T + 79T^{2} \)
83 \( 1 - 9.71T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
97 \( 1 + 2.27T + 97T^{2} \)
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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

−7.977024002775825407394885844123, −6.97036839466377128523483747732, −6.44884738916461701236393469618, −5.62665996993556968099151409526, −4.80689955632370036786704793773, −4.17600739261999026216324636018, −3.54070431498146186251073261221, −3.39640220590609993626265355560, −1.78275278529800304395848751802, −0.57036742020563122022466101262, 0.57036742020563122022466101262, 1.78275278529800304395848751802, 3.39640220590609993626265355560, 3.54070431498146186251073261221, 4.17600739261999026216324636018, 4.80689955632370036786704793773, 5.62665996993556968099151409526, 6.44884738916461701236393469618, 6.97036839466377128523483747732, 7.977024002775825407394885844123

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