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  − 1.56·2-s + 0.434·4-s + 1.69·5-s − 7-s + 2.44·8-s − 2.64·10-s + 1.57·11-s + 5.37·13-s + 1.56·14-s − 4.68·16-s − 7.38·17-s − 0.683·19-s + 0.737·20-s − 2.46·22-s + 8.53·23-s − 2.11·25-s − 8.38·26-s − 0.434·28-s − 5.81·29-s + 7.97·31-s + 2.41·32-s + 11.5·34-s − 1.69·35-s − 11.4·37-s + 1.06·38-s + 4.14·40-s − 5.44·41-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.217·4-s + 0.759·5-s − 0.377·7-s + 0.863·8-s − 0.837·10-s + 0.475·11-s + 1.49·13-s + 0.416·14-s − 1.17·16-s − 1.79·17-s − 0.156·19-s + 0.164·20-s − 0.524·22-s + 1.77·23-s − 0.423·25-s − 1.64·26-s − 0.0821·28-s − 1.08·29-s + 1.43·31-s + 0.427·32-s + 1.97·34-s − 0.287·35-s − 1.87·37-s + 0.173·38-s + 0.655·40-s − 0.849·41-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 \( 1 + 1.56T + 2T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 - 1.57T + 11T^{2} \)
13 \( 1 - 5.37T + 13T^{2} \)
17 \( 1 + 7.38T + 17T^{2} \)
19 \( 1 + 0.683T + 19T^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 + 5.81T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 - 4.36T + 43T^{2} \)
47 \( 1 - 3.11T + 47T^{2} \)
53 \( 1 + 7.03T + 53T^{2} \)
59 \( 1 - 5.82T + 59T^{2} \)
61 \( 1 + 6.57T + 61T^{2} \)
67 \( 1 + 6.68T + 67T^{2} \)
71 \( 1 + 4.90T + 71T^{2} \)
73 \( 1 + 7.91T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 6.39T + 83T^{2} \)
89 \( 1 + 2.66T + 89T^{2} \)
97 \( 1 - 5.37T + 97T^{2} \)
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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

−7.53550310675468913213219939302, −6.78720979801600672941958934112, −6.38955220944490650361070033159, −5.53822847508634122812984228637, −4.62227382912544251626944405556, −3.93160809650420639610281645816, −2.92550387835020930174111774169, −1.84309822709592688985233697409, −1.24810865734421170113780831839, 0, 1.24810865734421170113780831839, 1.84309822709592688985233697409, 2.92550387835020930174111774169, 3.93160809650420639610281645816, 4.62227382912544251626944405556, 5.53822847508634122812984228637, 6.38955220944490650361070033159, 6.78720979801600672941958934112, 7.53550310675468913213219939302

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