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  + 0.0366·2-s − 1.99·4-s − 1.02·5-s + 7-s − 0.146·8-s − 0.0374·10-s + 4.36·11-s − 3.87·13-s + 0.0366·14-s + 3.99·16-s − 6.07·17-s + 6.90·19-s + 2.04·20-s + 0.159·22-s + 5.17·23-s − 3.95·25-s − 0.141·26-s − 1.99·28-s + 1.01·29-s + 8.54·31-s + 0.439·32-s − 0.222·34-s − 1.02·35-s − 8.38·37-s + 0.252·38-s + 0.149·40-s − 1.56·41-s + ⋯
L(s)  = 1  + 0.0259·2-s − 0.999·4-s − 0.456·5-s + 0.377·7-s − 0.0518·8-s − 0.0118·10-s + 1.31·11-s − 1.07·13-s + 0.00979·14-s + 0.997·16-s − 1.47·17-s + 1.58·19-s + 0.456·20-s + 0.0341·22-s + 1.07·23-s − 0.791·25-s − 0.0278·26-s − 0.377·28-s + 0.187·29-s + 1.53·31-s + 0.0776·32-s − 0.0381·34-s − 0.172·35-s − 1.37·37-s + 0.0410·38-s + 0.0236·40-s − 0.244·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.347876975$
$L(\frac12)$  $\approx$  $1.347876975$
$L(\frac{3}{2})$   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 \( 1 - 0.0366T + 2T^{2} \)
5 \( 1 + 1.02T + 5T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 - 6.90T + 19T^{2} \)
23 \( 1 - 5.17T + 23T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 - 8.54T + 31T^{2} \)
37 \( 1 + 8.38T + 37T^{2} \)
41 \( 1 + 1.56T + 41T^{2} \)
43 \( 1 + 6.37T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 - 5.22T + 53T^{2} \)
59 \( 1 - 7.49T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 7.55T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 1.99T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 3.14T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−7.929504801904337636923417943601, −7.05148862469944154426678454051, −6.69353155632116630129755437617, −5.49654168415676593299687259288, −4.94879816984550298821973111345, −4.33138650575698229544428211091, −3.68712565782435139967184673174, −2.81681626500456084394691056854, −1.59661759672385827132254736492, −0.60365147971315631336388367902, 0.60365147971315631336388367902, 1.59661759672385827132254736492, 2.81681626500456084394691056854, 3.68712565782435139967184673174, 4.33138650575698229544428211091, 4.94879816984550298821973111345, 5.49654168415676593299687259288, 6.69353155632116630129755437617, 7.05148862469944154426678454051, 7.929504801904337636923417943601

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