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  + 1.91·2-s + 1.67·4-s − 1.63·5-s + 7-s − 0.619·8-s − 3.12·10-s − 4.53·11-s + 0.785·13-s + 1.91·14-s − 4.54·16-s + 4.58·17-s + 7.32·19-s − 2.73·20-s − 8.70·22-s − 1.77·23-s − 2.34·25-s + 1.50·26-s + 1.67·28-s − 4.77·29-s + 5.06·31-s − 7.47·32-s + 8.79·34-s − 1.63·35-s − 1.03·37-s + 14.0·38-s + 1.00·40-s + 10.6·41-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.838·4-s − 0.728·5-s + 0.377·7-s − 0.218·8-s − 0.988·10-s − 1.36·11-s + 0.217·13-s + 0.512·14-s − 1.13·16-s + 1.11·17-s + 1.67·19-s − 0.611·20-s − 1.85·22-s − 0.371·23-s − 0.468·25-s + 0.295·26-s + 0.316·28-s − 0.886·29-s + 0.909·31-s − 1.32·32-s + 1.50·34-s − 0.275·35-s − 0.169·37-s + 2.27·38-s + 0.159·40-s + 1.66·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.217807522$
$L(\frac12)$  $\approx$  $3.217807522$
$L(\frac{3}{2})$   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 \( 1 - 1.91T + 2T^{2} \)
5 \( 1 + 1.63T + 5T^{2} \)
11 \( 1 + 4.53T + 11T^{2} \)
13 \( 1 - 0.785T + 13T^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 - 7.32T + 19T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 + 4.77T + 29T^{2} \)
31 \( 1 - 5.06T + 31T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 8.26T + 43T^{2} \)
47 \( 1 + 7.59T + 47T^{2} \)
53 \( 1 + 0.151T + 53T^{2} \)
59 \( 1 + 4.69T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 1.56T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 5.59T + 73T^{2} \)
79 \( 1 + 3.81T + 79T^{2} \)
83 \( 1 + 0.734T + 83T^{2} \)
89 \( 1 - 5.16T + 89T^{2} \)
97 \( 1 + 5.35T + 97T^{2} \)
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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

−7.71220785867121610911799562961, −7.26854224217859601136167734349, −6.01914347447708797729859856256, −5.67401083319521331867317887290, −4.95422285501201317823145577218, −4.39761488444202683372117784410, −3.47195000083431428719839773328, −3.09764708421929988227127130649, −2.11999361976959258989592476997, −0.70691517819251952633299492725, 0.70691517819251952633299492725, 2.11999361976959258989592476997, 3.09764708421929988227127130649, 3.47195000083431428719839773328, 4.39761488444202683372117784410, 4.95422285501201317823145577218, 5.67401083319521331867317887290, 6.01914347447708797729859856256, 7.26854224217859601136167734349, 7.71220785867121610911799562961

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