Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s − 13-s + 2·14-s − 4·16-s − 6·17-s − 4·19-s − 2·20-s − 9·23-s − 4·25-s + 2·26-s − 2·28-s − 5·29-s − 3·31-s + 8·32-s + 12·34-s + 35-s + 37-s + 8·38-s − 10·41-s − 4·43-s + 18·46-s − 12·47-s + 49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s − 0.277·13-s + 0.534·14-s − 16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 1.87·23-s − 4/5·25-s + 0.392·26-s − 0.377·28-s − 0.928·29-s − 0.538·31-s + 1.41·32-s + 2.05·34-s + 0.169·35-s + 0.164·37-s + 1.29·38-s − 1.56·41-s − 0.609·43-s + 2.65·46-s − 1.75·47-s + 1/7·49-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  =  \(2\)
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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
127 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.38913883767735899260996621631, −6.59336509126242971767925344627, −6.16351773625788216164559558356, −4.96017372560659992918891266557, −4.21597762495776692575265182128, −3.49935483638696849760138329940, −2.17017920068452855217938294274, −1.78586874663305247485356312219, 0, 0, 1.78586874663305247485356312219, 2.17017920068452855217938294274, 3.49935483638696849760138329940, 4.21597762495776692575265182128, 4.96017372560659992918891266557, 6.16351773625788216164559558356, 6.59336509126242971767925344627, 7.38913883767735899260996621631

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