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  + 1.94·2-s + 1.77·4-s − 3.66·5-s + 7-s − 0.445·8-s − 7.11·10-s − 2.76·11-s − 3.90·13-s + 1.94·14-s − 4.40·16-s − 3.65·17-s − 2.93·19-s − 6.49·20-s − 5.36·22-s + 5.90·23-s + 8.44·25-s − 7.57·26-s + 1.77·28-s − 1.00·29-s − 0.885·31-s − 7.66·32-s − 7.09·34-s − 3.66·35-s + 5.97·37-s − 5.70·38-s + 1.63·40-s − 7.04·41-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.885·4-s − 1.63·5-s + 0.377·7-s − 0.157·8-s − 2.25·10-s − 0.833·11-s − 1.08·13-s + 0.518·14-s − 1.10·16-s − 0.886·17-s − 0.673·19-s − 1.45·20-s − 1.14·22-s + 1.23·23-s + 1.68·25-s − 1.48·26-s + 0.334·28-s − 0.187·29-s − 0.158·31-s − 1.35·32-s − 1.21·34-s − 0.619·35-s + 0.981·37-s − 0.924·38-s + 0.258·40-s − 1.09·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.640607611$
$L(\frac12)$  $\approx$  $1.640607611$
$L(\frac{3}{2})$   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 \( 1 - 1.94T + 2T^{2} \)
5 \( 1 + 3.66T + 5T^{2} \)
11 \( 1 + 2.76T + 11T^{2} \)
13 \( 1 + 3.90T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 + 0.885T + 31T^{2} \)
37 \( 1 - 5.97T + 37T^{2} \)
41 \( 1 + 7.04T + 41T^{2} \)
43 \( 1 - 5.95T + 43T^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 + 4.27T + 53T^{2} \)
59 \( 1 - 7.36T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 + 4.06T + 71T^{2} \)
73 \( 1 - 1.01T + 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 + 3.97T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.57415099767645425530090715996, −7.14657056841316009861308689329, −6.45061625141016088231972633509, −5.36855464243524206424478467446, −4.89682905328664486872609027469, −4.32342296283999083550607875084, −3.77022871821564165801963028420, −2.86447757080285801276647311951, −2.29360290234172728368705552752, −0.48140514721191797816145867654, 0.48140514721191797816145867654, 2.29360290234172728368705552752, 2.86447757080285801276647311951, 3.77022871821564165801963028420, 4.32342296283999083550607875084, 4.89682905328664486872609027469, 5.36855464243524206424478467446, 6.45061625141016088231972633509, 7.14657056841316009861308689329, 7.57415099767645425530090715996

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