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  + 0.249·2-s − 1.93·4-s − 0.989·5-s + 7-s − 0.981·8-s − 0.246·10-s + 2.01·11-s + 4.18·13-s + 0.249·14-s + 3.63·16-s + 7.51·17-s + 7.64·19-s + 1.91·20-s + 0.503·22-s + 2.82·23-s − 4.02·25-s + 1.04·26-s − 1.93·28-s − 5.08·29-s + 1.14·31-s + 2.86·32-s + 1.87·34-s − 0.989·35-s + 10.6·37-s + 1.90·38-s + 0.971·40-s − 8.51·41-s + ⋯
L(s)  = 1  + 0.176·2-s − 0.968·4-s − 0.442·5-s + 0.377·7-s − 0.346·8-s − 0.0779·10-s + 0.608·11-s + 1.15·13-s + 0.0665·14-s + 0.907·16-s + 1.82·17-s + 1.75·19-s + 0.428·20-s + 0.107·22-s + 0.589·23-s − 0.804·25-s + 0.204·26-s − 0.366·28-s − 0.944·29-s + 0.205·31-s + 0.506·32-s + 0.320·34-s − 0.167·35-s + 1.74·37-s + 0.309·38-s + 0.153·40-s − 1.33·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.184952272$
$L(\frac12)$  $\approx$  $2.184952272$
$L(\frac{3}{2})$   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 \( 1 - 0.249T + 2T^{2} \)
5 \( 1 + 0.989T + 5T^{2} \)
11 \( 1 - 2.01T + 11T^{2} \)
13 \( 1 - 4.18T + 13T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 - 7.64T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 5.08T + 29T^{2} \)
31 \( 1 - 1.14T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 8.51T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 - 0.347T + 47T^{2} \)
53 \( 1 - 4.33T + 53T^{2} \)
59 \( 1 - 9.40T + 59T^{2} \)
61 \( 1 + 3.40T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−7.889592633890453672732673827384, −7.37037581375330684180995042443, −6.31211944004051909252853694082, −5.49507595730850713900346944675, −5.21515407029071671168071637393, −4.09778166740747395470103818692, −3.65451532446080884738010189165, −3.04372378686940196456665110576, −1.41449337677926837026385971292, −0.834082657759181678977829722454, 0.834082657759181678977829722454, 1.41449337677926837026385971292, 3.04372378686940196456665110576, 3.65451532446080884738010189165, 4.09778166740747395470103818692, 5.21515407029071671168071637393, 5.49507595730850713900346944675, 6.31211944004051909252853694082, 7.37037581375330684180995042443, 7.889592633890453672732673827384

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