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  + 0.942·2-s − 1.11·4-s + 2.02·5-s − 7-s − 2.93·8-s + 1.91·10-s + 0.393·11-s − 1.52·13-s − 0.942·14-s − 0.541·16-s − 7.22·17-s + 5.33·19-s − 2.25·20-s + 0.371·22-s + 6.14·23-s − 0.889·25-s − 1.43·26-s + 1.11·28-s + 10.3·29-s − 5.51·31-s + 5.35·32-s − 6.81·34-s − 2.02·35-s + 1.11·37-s + 5.02·38-s − 5.94·40-s − 12.0·41-s + ⋯
L(s)  = 1  + 0.666·2-s − 0.555·4-s + 0.906·5-s − 0.377·7-s − 1.03·8-s + 0.604·10-s + 0.118·11-s − 0.422·13-s − 0.251·14-s − 0.135·16-s − 1.75·17-s + 1.22·19-s − 0.503·20-s + 0.0791·22-s + 1.28·23-s − 0.177·25-s − 0.281·26-s + 0.210·28-s + 1.92·29-s − 0.989·31-s + 0.946·32-s − 1.16·34-s − 0.342·35-s + 0.183·37-s + 0.815·38-s − 0.940·40-s − 1.87·41-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 \( 1 - 0.942T + 2T^{2} \)
5 \( 1 - 2.02T + 5T^{2} \)
11 \( 1 - 0.393T + 11T^{2} \)
13 \( 1 + 1.52T + 13T^{2} \)
17 \( 1 + 7.22T + 17T^{2} \)
19 \( 1 - 5.33T + 19T^{2} \)
23 \( 1 - 6.14T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 5.51T + 31T^{2} \)
37 \( 1 - 1.11T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 0.215T + 43T^{2} \)
47 \( 1 + 4.05T + 47T^{2} \)
53 \( 1 - 6.05T + 53T^{2} \)
59 \( 1 - 3.92T + 59T^{2} \)
61 \( 1 + 3.48T + 61T^{2} \)
67 \( 1 - 9.31T + 67T^{2} \)
71 \( 1 + 6.78T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.18167703452706414880590645957, −6.70148214511896257904336367951, −6.00806572321309297073191533607, −5.22666913757350650411301734446, −4.83744614087059475772723955792, −3.99582352236972075329758635495, −3.08532922540225097630758052264, −2.50043358675860376995975355482, −1.31848185598751576174362142076, 0, 1.31848185598751576174362142076, 2.50043358675860376995975355482, 3.08532922540225097630758052264, 3.99582352236972075329758635495, 4.83744614087059475772723955792, 5.22666913757350650411301734446, 6.00806572321309297073191533607, 6.70148214511896257904336367951, 7.18167703452706414880590645957

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