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  − 2.55·2-s + 4.52·4-s + 2.12·5-s + 7-s − 6.45·8-s − 5.43·10-s + 4.13·11-s − 4.75·13-s − 2.55·14-s + 7.43·16-s + 1.32·17-s − 5.03·19-s + 9.63·20-s − 10.5·22-s − 0.634·23-s − 0.468·25-s + 12.1·26-s + 4.52·28-s + 6.95·29-s − 3.31·31-s − 6.08·32-s − 3.38·34-s + 2.12·35-s + 7.32·37-s + 12.8·38-s − 13.7·40-s − 4.92·41-s + ⋯
L(s)  = 1  − 1.80·2-s + 2.26·4-s + 0.951·5-s + 0.377·7-s − 2.28·8-s − 1.71·10-s + 1.24·11-s − 1.31·13-s − 0.682·14-s + 1.85·16-s + 0.321·17-s − 1.15·19-s + 2.15·20-s − 2.25·22-s − 0.132·23-s − 0.0937·25-s + 2.38·26-s + 0.855·28-s + 1.29·29-s − 0.595·31-s − 1.07·32-s − 0.580·34-s + 0.359·35-s + 1.20·37-s + 2.08·38-s − 2.17·40-s − 0.769·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.012792204$
$L(\frac12)$  $\approx$  $1.012792204$
$L(\frac{3}{2})$   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 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 - 2.12T + 5T^{2} \)
11 \( 1 - 4.13T + 11T^{2} \)
13 \( 1 + 4.75T + 13T^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 + 5.03T + 19T^{2} \)
23 \( 1 + 0.634T + 23T^{2} \)
29 \( 1 - 6.95T + 29T^{2} \)
31 \( 1 + 3.31T + 31T^{2} \)
37 \( 1 - 7.32T + 37T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 4.29T + 53T^{2} \)
59 \( 1 - 9.53T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 2.95T + 67T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 - 7.08T + 73T^{2} \)
79 \( 1 - 5.36T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 3.15T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−8.001065207641079258998451252821, −7.31722153622667786071362931132, −6.45273942683915990022879950529, −6.33344981066447811029139496921, −5.19282733573135420122414701757, −4.33128885073989600283214239449, −3.07489091571404183456914712867, −2.11181188428140247732831450744, −1.73115982534605602022777106803, −0.65761626822329944539557250662, 0.65761626822329944539557250662, 1.73115982534605602022777106803, 2.11181188428140247732831450744, 3.07489091571404183456914712867, 4.33128885073989600283214239449, 5.19282733573135420122414701757, 6.33344981066447811029139496921, 6.45273942683915990022879950529, 7.31722153622667786071362931132, 8.001065207641079258998451252821

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