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  − 1.90·2-s + 1.63·4-s − 1.97·5-s + 7-s + 0.698·8-s + 3.77·10-s + 5.17·11-s − 1.58·13-s − 1.90·14-s − 4.59·16-s + 8.13·17-s + 4.36·19-s − 3.23·20-s − 9.87·22-s + 7.89·23-s − 1.08·25-s + 3.02·26-s + 1.63·28-s − 2.09·29-s − 0.621·31-s + 7.36·32-s − 15.5·34-s − 1.97·35-s + 0.803·37-s − 8.31·38-s − 1.38·40-s + 2.54·41-s + ⋯
L(s)  = 1  − 1.34·2-s + 0.816·4-s − 0.885·5-s + 0.377·7-s + 0.247·8-s + 1.19·10-s + 1.56·11-s − 0.440·13-s − 0.509·14-s − 1.14·16-s + 1.97·17-s + 1.00·19-s − 0.723·20-s − 2.10·22-s + 1.64·23-s − 0.216·25-s + 0.593·26-s + 0.308·28-s − 0.388·29-s − 0.111·31-s + 1.30·32-s − 2.65·34-s − 0.334·35-s + 0.132·37-s − 1.34·38-s − 0.218·40-s + 0.397·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.235405056$
$L(\frac12)$  $\approx$  $1.235405056$
$L(\frac{3}{2})$   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 \( 1 + 1.90T + 2T^{2} \)
5 \( 1 + 1.97T + 5T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 - 8.13T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 - 7.89T + 23T^{2} \)
29 \( 1 + 2.09T + 29T^{2} \)
31 \( 1 + 0.621T + 31T^{2} \)
37 \( 1 - 0.803T + 37T^{2} \)
41 \( 1 - 2.54T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 5.22T + 59T^{2} \)
61 \( 1 - 5.63T + 61T^{2} \)
67 \( 1 - 7.63T + 67T^{2} \)
71 \( 1 - 2.26T + 71T^{2} \)
73 \( 1 - 3.81T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 1.82T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.77570277334650306039558389578, −7.38704975149987825090071664936, −6.96711765902769652844073019730, −5.81259791079281358941204979125, −5.08181941011072507933544707084, −4.09818286259246100382186545021, −3.57813382211351059706159127834, −2.47018141019559096380149782930, −1.13653292908963729247847794266, −0.897676057611291859060084143837, 0.897676057611291859060084143837, 1.13653292908963729247847794266, 2.47018141019559096380149782930, 3.57813382211351059706159127834, 4.09818286259246100382186545021, 5.08181941011072507933544707084, 5.81259791079281358941204979125, 6.96711765902769652844073019730, 7.38704975149987825090071664936, 7.77570277334650306039558389578

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