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  − 0.307·2-s − 1.90·4-s + 0.988·5-s + 7-s + 1.20·8-s − 0.303·10-s − 2.00·11-s − 0.289·13-s − 0.307·14-s + 3.44·16-s − 0.186·17-s + 0.852·19-s − 1.88·20-s + 0.616·22-s + 3.33·23-s − 4.02·25-s + 0.0890·26-s − 1.90·28-s + 4.37·29-s − 5.73·31-s − 3.45·32-s + 0.0572·34-s + 0.988·35-s − 7.05·37-s − 0.261·38-s + 1.18·40-s − 3.81·41-s + ⋯
L(s)  = 1  − 0.217·2-s − 0.952·4-s + 0.442·5-s + 0.377·7-s + 0.424·8-s − 0.0960·10-s − 0.605·11-s − 0.0803·13-s − 0.0821·14-s + 0.860·16-s − 0.0451·17-s + 0.195·19-s − 0.421·20-s + 0.131·22-s + 0.695·23-s − 0.804·25-s + 0.0174·26-s − 0.360·28-s + 0.812·29-s − 1.03·31-s − 0.611·32-s + 0.00981·34-s + 0.167·35-s − 1.15·37-s − 0.0424·38-s + 0.187·40-s − 0.596·41-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 \( 1 + 0.307T + 2T^{2} \)
5 \( 1 - 0.988T + 5T^{2} \)
11 \( 1 + 2.00T + 11T^{2} \)
13 \( 1 + 0.289T + 13T^{2} \)
17 \( 1 + 0.186T + 17T^{2} \)
19 \( 1 - 0.852T + 19T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
37 \( 1 + 7.05T + 37T^{2} \)
41 \( 1 + 3.81T + 41T^{2} \)
43 \( 1 - 9.06T + 43T^{2} \)
47 \( 1 + 1.34T + 47T^{2} \)
53 \( 1 + 3.75T + 53T^{2} \)
59 \( 1 - 4.26T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 0.526T + 89T^{2} \)
97 \( 1 - 6.83T + 97T^{2} \)
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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

−7.61101355213255224160276775080, −6.93809730268895159346427513449, −5.91439686074406098642649852177, −5.32392641924756193435717652293, −4.79027192212279781619500303321, −3.97196051223100766481451386034, −3.14473485645889243656324651941, −2.11916747369891743343312801012, −1.18339700259551294861121555385, 0, 1.18339700259551294861121555385, 2.11916747369891743343312801012, 3.14473485645889243656324651941, 3.97196051223100766481451386034, 4.79027192212279781619500303321, 5.32392641924756193435717652293, 5.91439686074406098642649852177, 6.93809730268895159346427513449, 7.61101355213255224160276775080

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