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  − 0.959·2-s − 1.07·4-s − 2.40·5-s − 7-s + 2.95·8-s + 2.30·10-s − 4.25·11-s + 3.54·13-s + 0.959·14-s − 0.679·16-s − 0.302·17-s − 6.75·19-s + 2.59·20-s + 4.08·22-s + 6.51·23-s + 0.782·25-s − 3.40·26-s + 1.07·28-s + 3.90·29-s − 5.48·31-s − 5.25·32-s + 0.290·34-s + 2.40·35-s − 6.55·37-s + 6.48·38-s − 7.10·40-s + 2.17·41-s + ⋯
L(s)  = 1  − 0.678·2-s − 0.539·4-s − 1.07·5-s − 0.377·7-s + 1.04·8-s + 0.729·10-s − 1.28·11-s + 0.982·13-s + 0.256·14-s − 0.169·16-s − 0.0733·17-s − 1.55·19-s + 0.579·20-s + 0.871·22-s + 1.35·23-s + 0.156·25-s − 0.667·26-s + 0.203·28-s + 0.725·29-s − 0.985·31-s − 0.929·32-s + 0.0497·34-s + 0.406·35-s − 1.07·37-s + 1.05·38-s − 1.12·40-s + 0.339·41-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 \( 1 + 0.959T + 2T^{2} \)
5 \( 1 + 2.40T + 5T^{2} \)
11 \( 1 + 4.25T + 11T^{2} \)
13 \( 1 - 3.54T + 13T^{2} \)
17 \( 1 + 0.302T + 17T^{2} \)
19 \( 1 + 6.75T + 19T^{2} \)
23 \( 1 - 6.51T + 23T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 + 5.48T + 31T^{2} \)
37 \( 1 + 6.55T + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 9.47T + 43T^{2} \)
47 \( 1 - 1.43T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 - 1.05T + 61T^{2} \)
67 \( 1 + 4.72T + 67T^{2} \)
71 \( 1 - 6.25T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 4.55T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 - 1.74T + 97T^{2} \)
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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

−7.59970482167619005118940600617, −7.12616224287042355910778609578, −6.17027274396159190964288497286, −5.32126214012925169024880773875, −4.55622309592903232639882635400, −3.93909388895491173980303612203, −3.20728842791737800991915013102, −2.14174786823942383315208645829, −0.847466171499621743089099854139, 0, 0.847466171499621743089099854139, 2.14174786823942383315208645829, 3.20728842791737800991915013102, 3.93909388895491173980303612203, 4.55622309592903232639882635400, 5.32126214012925169024880773875, 6.17027274396159190964288497286, 7.12616224287042355910778609578, 7.59970482167619005118940600617

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