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  − 2.28·2-s + 3.21·4-s − 2.58·5-s + 7-s − 2.77·8-s + 5.90·10-s − 3.21·11-s − 1.17·13-s − 2.28·14-s − 0.100·16-s − 1.07·17-s + 5.70·19-s − 8.31·20-s + 7.33·22-s + 0.390·23-s + 1.68·25-s + 2.68·26-s + 3.21·28-s + 1.39·29-s − 0.821·31-s + 5.77·32-s + 2.44·34-s − 2.58·35-s + 2.05·37-s − 13.0·38-s + 7.16·40-s + 0.263·41-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.60·4-s − 1.15·5-s + 0.377·7-s − 0.979·8-s + 1.86·10-s − 0.969·11-s − 0.326·13-s − 0.610·14-s − 0.0252·16-s − 0.259·17-s + 1.30·19-s − 1.85·20-s + 1.56·22-s + 0.0814·23-s + 0.337·25-s + 0.527·26-s + 0.607·28-s + 0.259·29-s − 0.147·31-s + 1.02·32-s + 0.419·34-s − 0.437·35-s + 0.337·37-s − 2.11·38-s + 1.13·40-s + 0.0411·41-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 \( 1 + 2.28T + 2T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + 1.07T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 - 0.390T + 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 + 0.821T + 31T^{2} \)
37 \( 1 - 2.05T + 37T^{2} \)
41 \( 1 - 0.263T + 41T^{2} \)
43 \( 1 + 8.34T + 43T^{2} \)
47 \( 1 + 8.74T + 47T^{2} \)
53 \( 1 + 0.256T + 53T^{2} \)
59 \( 1 - 5.69T + 59T^{2} \)
61 \( 1 + 2.53T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 9.20T + 73T^{2} \)
79 \( 1 - 0.721T + 79T^{2} \)
83 \( 1 - 5.52T + 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−7.76224295216424980226764394811, −7.16339434831929839021267792987, −6.55153621887667465670564534496, −5.33248738865637886256338383170, −4.77515857594000585840209564764, −3.73244107007387318403567008748, −2.87683779500171998023402067744, −1.98287533180890307041429277163, −0.897165543011092017387848736399, 0, 0.897165543011092017387848736399, 1.98287533180890307041429277163, 2.87683779500171998023402067744, 3.73244107007387318403567008748, 4.77515857594000585840209564764, 5.33248738865637886256338383170, 6.55153621887667465670564534496, 7.16339434831929839021267792987, 7.76224295216424980226764394811

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