Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.49·2-s + 0.220·4-s − 2.04·5-s + 7-s + 2.65·8-s + 3.05·10-s + 1.77·11-s − 4.05·13-s − 1.49·14-s − 4.39·16-s − 7.64·17-s + 3.24·19-s − 0.452·20-s − 2.64·22-s − 6.77·23-s − 0.807·25-s + 6.04·26-s + 0.220·28-s − 6.90·29-s − 1.66·31-s + 1.24·32-s + 11.3·34-s − 2.04·35-s + 9.03·37-s − 4.84·38-s − 5.42·40-s − 9.40·41-s + ⋯
L(s)  = 1  − 1.05·2-s + 0.110·4-s − 0.915·5-s + 0.377·7-s + 0.937·8-s + 0.964·10-s + 0.535·11-s − 1.12·13-s − 0.398·14-s − 1.09·16-s − 1.85·17-s + 0.745·19-s − 0.101·20-s − 0.564·22-s − 1.41·23-s − 0.161·25-s + 1.18·26-s + 0.0417·28-s − 1.28·29-s − 0.298·31-s + 0.219·32-s + 1.95·34-s − 0.346·35-s + 1.48·37-s − 0.785·38-s − 0.858·40-s − 1.46·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2697651382$
$L(\frac12)$  $\approx$  $0.2697651382$
$L(\frac{3}{2})$   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 \( 1 + 1.49T + 2T^{2} \)
5 \( 1 + 2.04T + 5T^{2} \)
11 \( 1 - 1.77T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 + 7.64T + 17T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 + 6.77T + 23T^{2} \)
29 \( 1 + 6.90T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 - 9.03T + 37T^{2} \)
41 \( 1 + 9.40T + 41T^{2} \)
43 \( 1 - 2.27T + 43T^{2} \)
47 \( 1 - 1.14T + 47T^{2} \)
53 \( 1 + 0.909T + 53T^{2} \)
59 \( 1 - 0.510T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 0.730T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 - 3.42T + 73T^{2} \)
79 \( 1 + 7.04T + 79T^{2} \)
83 \( 1 + 1.32T + 83T^{2} \)
89 \( 1 - 3.63T + 89T^{2} \)
97 \( 1 + 9.35T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76822740519952951299905248464, −7.50185682805880376645243333934, −6.78849767982248221677491529182, −5.84989499749629417424371868286, −4.77775459682507068345712285054, −4.35277971855172320747583361835, −3.65294995843675811061864788957, −2.34775380951657980026163975212, −1.63772333162806729280895591262, −0.30111629331361084461420006685, 0.30111629331361084461420006685, 1.63772333162806729280895591262, 2.34775380951657980026163975212, 3.65294995843675811061864788957, 4.35277971855172320747583361835, 4.77775459682507068345712285054, 5.84989499749629417424371868286, 6.78849767982248221677491529182, 7.50185682805880376645243333934, 7.76822740519952951299905248464

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