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  + 1.60·2-s + 0.579·4-s + 3.54·5-s + 7-s − 2.28·8-s + 5.69·10-s + 2.78·11-s + 2.43·13-s + 1.60·14-s − 4.82·16-s + 6.06·17-s − 1.93·19-s + 2.05·20-s + 4.47·22-s − 3.19·23-s + 7.58·25-s + 3.91·26-s + 0.579·28-s − 2.42·29-s + 3.23·31-s − 3.18·32-s + 9.73·34-s + 3.54·35-s − 1.06·37-s − 3.11·38-s − 8.09·40-s + 3.51·41-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.289·4-s + 1.58·5-s + 0.377·7-s − 0.806·8-s + 1.80·10-s + 0.839·11-s + 0.675·13-s + 0.429·14-s − 1.20·16-s + 1.46·17-s − 0.444·19-s + 0.460·20-s + 0.953·22-s − 0.666·23-s + 1.51·25-s + 0.767·26-s + 0.109·28-s − 0.450·29-s + 0.581·31-s − 0.563·32-s + 1.66·34-s + 0.599·35-s − 0.175·37-s − 0.504·38-s − 1.27·40-s + 0.548·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.626192996$
$L(\frac12)$  $\approx$  $5.626192996$
$L(\frac{3}{2})$   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 \( 1 - 1.60T + 2T^{2} \)
5 \( 1 - 3.54T + 5T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 2.43T + 13T^{2} \)
17 \( 1 - 6.06T + 17T^{2} \)
19 \( 1 + 1.93T + 19T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 + 2.42T + 29T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 + 1.06T + 37T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 5.82T + 43T^{2} \)
47 \( 1 - 2.39T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 2.80T + 59T^{2} \)
61 \( 1 - 8.16T + 61T^{2} \)
67 \( 1 - 5.97T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 2.47T + 73T^{2} \)
79 \( 1 + 3.02T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 2.30T + 89T^{2} \)
97 \( 1 + 0.707T + 97T^{2} \)
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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

−7.79598193708336686342785887679, −6.72099804404728436332362112704, −6.16430593329467488067696575834, −5.71479250976922685451498545779, −5.18555275129482984466881784607, −4.29851246700164764283426762766, −3.65440438945229270624810254666, −2.78484637824462003689728704760, −1.92263857776297973447264644984, −1.07039467205228159817059620562, 1.07039467205228159817059620562, 1.92263857776297973447264644984, 2.78484637824462003689728704760, 3.65440438945229270624810254666, 4.29851246700164764283426762766, 5.18555275129482984466881784607, 5.71479250976922685451498545779, 6.16430593329467488067696575834, 6.72099804404728436332362112704, 7.79598193708336686342785887679

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