Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·2-s − 0.378·4-s + 2.08·5-s + 7-s + 3.02·8-s − 2.65·10-s + 2.11·11-s − 3.46·13-s − 1.27·14-s − 3.10·16-s − 5.03·17-s + 0.224·19-s − 0.789·20-s − 2.68·22-s + 7.75·23-s − 0.649·25-s + 4.41·26-s − 0.378·28-s − 5.57·29-s − 0.678·31-s − 2.10·32-s + 6.40·34-s + 2.08·35-s + 10.9·37-s − 0.285·38-s + 6.31·40-s − 1.33·41-s + ⋯
L(s)  = 1  − 0.900·2-s − 0.189·4-s + 0.932·5-s + 0.377·7-s + 1.07·8-s − 0.839·10-s + 0.636·11-s − 0.961·13-s − 0.340·14-s − 0.775·16-s − 1.22·17-s + 0.0514·19-s − 0.176·20-s − 0.573·22-s + 1.61·23-s − 0.129·25-s + 0.865·26-s − 0.0715·28-s − 1.03·29-s − 0.121·31-s − 0.372·32-s + 1.09·34-s + 0.352·35-s + 1.79·37-s − 0.0463·38-s + 0.998·40-s − 0.208·41-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  =  1
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.27T + 2T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + 5.03T + 17T^{2} \)
19 \( 1 - 0.224T + 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 + 0.678T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 1.33T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + 0.612T + 47T^{2} \)
53 \( 1 + 9.19T + 53T^{2} \)
59 \( 1 + 9.06T + 59T^{2} \)
61 \( 1 - 0.604T + 61T^{2} \)
67 \( 1 + 1.65T + 67T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + 7.49T + 73T^{2} \)
79 \( 1 - 5.41T + 79T^{2} \)
83 \( 1 + 5.01T + 83T^{2} \)
89 \( 1 + 8.62T + 89T^{2} \)
97 \( 1 - 6.43T + 97T^{2} \)
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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

−7.57917171513614152772447347951, −6.93185740028706589358184296878, −6.26777603792509147953792528764, −5.30442201848888592656105360103, −4.74950625709710896797072467422, −4.06710667206238230247728499201, −2.82187072967682952640315008712, −1.94745489021789217979004090629, −1.26524132231775778186577920827, 0, 1.26524132231775778186577920827, 1.94745489021789217979004090629, 2.82187072967682952640315008712, 4.06710667206238230247728499201, 4.74950625709710896797072467422, 5.30442201848888592656105360103, 6.26777603792509147953792528764, 6.93185740028706589358184296878, 7.57917171513614152772447347951

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