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  + 1.58·2-s + 0.523·4-s − 3.57·5-s + 7-s − 2.34·8-s − 5.68·10-s + 2.48·11-s + 0.477·13-s + 1.58·14-s − 4.77·16-s + 2.92·17-s − 1.96·19-s − 1.87·20-s + 3.94·22-s + 1.07·23-s + 7.79·25-s + 0.758·26-s + 0.523·28-s − 5.99·29-s + 6.77·31-s − 2.89·32-s + 4.65·34-s − 3.57·35-s + 4.95·37-s − 3.11·38-s + 8.38·40-s + 2.60·41-s + ⋯
L(s)  = 1  + 1.12·2-s + 0.261·4-s − 1.59·5-s + 0.377·7-s − 0.829·8-s − 1.79·10-s + 0.749·11-s + 0.132·13-s + 0.424·14-s − 1.19·16-s + 0.709·17-s − 0.450·19-s − 0.418·20-s + 0.841·22-s + 0.225·23-s + 1.55·25-s + 0.148·26-s + 0.0989·28-s − 1.11·29-s + 1.21·31-s − 0.511·32-s + 0.797·34-s − 0.604·35-s + 0.815·37-s − 0.506·38-s + 1.32·40-s + 0.407·41-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 \( 1 - 1.58T + 2T^{2} \)
5 \( 1 + 3.57T + 5T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 - 0.477T + 13T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 + 1.96T + 19T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 4.95T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 0.254T + 53T^{2} \)
59 \( 1 - 9.98T + 59T^{2} \)
61 \( 1 - 8.89T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 4.92T + 71T^{2} \)
73 \( 1 - 1.24T + 73T^{2} \)
79 \( 1 + 7.06T + 79T^{2} \)
83 \( 1 + 6.48T + 83T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + 8.21T + 97T^{2} \)
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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

−7.40919190074128822514577959130, −6.73042904974349605765566972735, −6.00286357740544826068214769967, −5.18068537361612727100631810203, −4.47756577336640265196854576134, −3.99461622075517335406094995559, −3.44841774635875315802642327405, −2.64984400710367620552851078525, −1.22167274262807304499687207290, 0, 1.22167274262807304499687207290, 2.64984400710367620552851078525, 3.44841774635875315802642327405, 3.99461622075517335406094995559, 4.47756577336640265196854576134, 5.18068537361612727100631810203, 6.00286357740544826068214769967, 6.73042904974349605765566972735, 7.40919190074128822514577959130

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