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  + 0.803·2-s − 1.35·4-s − 0.343·5-s + 7-s − 2.69·8-s − 0.276·10-s − 4.04·11-s − 5.07·13-s + 0.803·14-s + 0.539·16-s + 3.48·17-s − 2.23·19-s + 0.464·20-s − 3.25·22-s + 2.35·23-s − 4.88·25-s − 4.08·26-s − 1.35·28-s − 9.24·29-s + 5.54·31-s + 5.82·32-s + 2.79·34-s − 0.343·35-s + 5.87·37-s − 1.79·38-s + 0.925·40-s + 1.66·41-s + ⋯
L(s)  = 1  + 0.568·2-s − 0.676·4-s − 0.153·5-s + 0.377·7-s − 0.953·8-s − 0.0872·10-s − 1.21·11-s − 1.40·13-s + 0.214·14-s + 0.134·16-s + 0.844·17-s − 0.513·19-s + 0.103·20-s − 0.693·22-s + 0.491·23-s − 0.976·25-s − 0.800·26-s − 0.255·28-s − 1.71·29-s + 0.995·31-s + 1.02·32-s + 0.480·34-s − 0.0580·35-s + 0.965·37-s − 0.291·38-s + 0.146·40-s + 0.260·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.075643947$
$L(\frac12)$  $\approx$  $1.075643947$
$L(\frac{3}{2})$   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 \( 1 - 0.803T + 2T^{2} \)
5 \( 1 + 0.343T + 5T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 3.48T + 17T^{2} \)
19 \( 1 + 2.23T + 19T^{2} \)
23 \( 1 - 2.35T + 23T^{2} \)
29 \( 1 + 9.24T + 29T^{2} \)
31 \( 1 - 5.54T + 31T^{2} \)
37 \( 1 - 5.87T + 37T^{2} \)
41 \( 1 - 1.66T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 4.97T + 67T^{2} \)
71 \( 1 + 8.55T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 0.366T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−7.77650483064548662072284473842, −7.38911696617409871829343108992, −6.17560508617870957268252040765, −5.61390006992664457835025723900, −4.86565214209283097243490273838, −4.56258677291893932631503542469, −3.53727429254859349624854640236, −2.83862721320760405801537916712, −1.95902264456525482511793745042, −0.45114137308278741250088895921, 0.45114137308278741250088895921, 1.95902264456525482511793745042, 2.83862721320760405801537916712, 3.53727429254859349624854640236, 4.56258677291893932631503542469, 4.86565214209283097243490273838, 5.61390006992664457835025723900, 6.17560508617870957268252040765, 7.38911696617409871829343108992, 7.77650483064548662072284473842

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