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  + 0.782·2-s − 1.38·4-s + 2.85·5-s + 7-s − 2.65·8-s + 2.23·10-s + 5.95·11-s − 2.24·13-s + 0.782·14-s + 0.702·16-s − 0.355·17-s + 4.58·19-s − 3.96·20-s + 4.65·22-s − 1.77·23-s + 3.15·25-s − 1.75·26-s − 1.38·28-s + 9.18·29-s − 2.61·31-s + 5.85·32-s − 0.277·34-s + 2.85·35-s + 0.446·37-s + 3.58·38-s − 7.57·40-s + 5.55·41-s + ⋯
L(s)  = 1  + 0.553·2-s − 0.693·4-s + 1.27·5-s + 0.377·7-s − 0.937·8-s + 0.706·10-s + 1.79·11-s − 0.622·13-s + 0.209·14-s + 0.175·16-s − 0.0861·17-s + 1.05·19-s − 0.886·20-s + 0.992·22-s − 0.369·23-s + 0.631·25-s − 0.344·26-s − 0.262·28-s + 1.70·29-s − 0.470·31-s + 1.03·32-s − 0.0476·34-s + 0.482·35-s + 0.0733·37-s + 0.581·38-s − 1.19·40-s + 0.867·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.470509064$
$L(\frac12)$  $\approx$  $3.470509064$
$L(\frac{3}{2})$   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 \( 1 - 0.782T + 2T^{2} \)
5 \( 1 - 2.85T + 5T^{2} \)
11 \( 1 - 5.95T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 + 0.355T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 - 9.18T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 - 0.446T + 37T^{2} \)
41 \( 1 - 5.55T + 41T^{2} \)
43 \( 1 + 3.81T + 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 1.31T + 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + 3.10T + 97T^{2} \)
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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

−7.85618614871023753635834776479, −6.90992240527028914760215667814, −6.24787115502401918449277532790, −5.76206398085447728489872815724, −4.96186806640141548118777683673, −4.44850851711856005433288831091, −3.60496948952144142748088975025, −2.76997627029649222235061726926, −1.74300312463644886952661318779, −0.914498046050812368305818817012, 0.914498046050812368305818817012, 1.74300312463644886952661318779, 2.76997627029649222235061726926, 3.60496948952144142748088975025, 4.44850851711856005433288831091, 4.96186806640141548118777683673, 5.76206398085447728489872815724, 6.24787115502401918449277532790, 6.90992240527028914760215667814, 7.85618614871023753635834776479

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