Properties

Degree $2$
Conductor $8001$
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  − 2.44·2-s + 3.99·4-s − 2.44·5-s + 7-s − 4.89·8-s + 5.99·10-s − 2.89·13-s − 2.44·14-s + 3.99·16-s + 5.44·17-s + 4.44·19-s − 9.79·20-s − 6·23-s + 0.999·25-s + 7.10·26-s + 3.99·28-s + 7.89·29-s + 8·31-s − 13.3·34-s − 2.44·35-s − 7·37-s − 10.8·38-s + 11.9·40-s − 5.44·41-s + 2·43-s + 14.6·46-s + 12·47-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.99·4-s − 1.09·5-s + 0.377·7-s − 1.73·8-s + 1.89·10-s − 0.804·13-s − 0.654·14-s + 0.999·16-s + 1.32·17-s + 1.02·19-s − 2.19·20-s − 1.25·23-s + 0.199·25-s + 1.39·26-s + 0.755·28-s + 1.46·29-s + 1.43·31-s − 2.28·34-s − 0.414·35-s − 1.15·37-s − 1.76·38-s + 1.89·40-s − 0.851·41-s + 0.304·43-s + 2.16·46-s + 1.75·47-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

Degree: \(2\)
Conductor: \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8001} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6138383216\)
\(L(\frac12)\) \(\approx\) \(0.6138383216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 - T \)
good2 \( 1 + 2.44T + 2T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 - 5.44T + 17T^{2} \)
19 \( 1 - 4.44T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 - 2.44T + 59T^{2} \)
61 \( 1 - 4.44T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 3.55T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 - 3.89T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 9.79T + 89T^{2} \)
97 \( 1 + 2.34T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.011752777519427564079207414797, −7.43587425908044755501057476873, −6.93245116135414882076674201949, −5.99581673424060231592990485831, −5.08630175609699120559103937533, −4.21167849494724243594894881229, −3.24591463955147718577444817011, −2.46312038230056082614469831264, −1.36281015229573483665711493292, −0.55150585258363356747223626786, 0.55150585258363356747223626786, 1.36281015229573483665711493292, 2.46312038230056082614469831264, 3.24591463955147718577444817011, 4.21167849494724243594894881229, 5.08630175609699120559103937533, 5.99581673424060231592990485831, 6.93245116135414882076674201949, 7.43587425908044755501057476873, 8.011752777519427564079207414797

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