Properties

Label 2-2001-1.1-c1-0-64
Degree $2$
Conductor $2001$
Sign $-1$
Analytic cond. $15.9780$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·2-s + 3-s + 3.63·4-s − 1.31·5-s − 2.37·6-s − 0.859·7-s − 3.87·8-s + 9-s + 3.12·10-s + 4.30·11-s + 3.63·12-s − 4.15·13-s + 2.03·14-s − 1.31·15-s + 1.93·16-s + 2.08·17-s − 2.37·18-s − 2.45·19-s − 4.78·20-s − 0.859·21-s − 10.2·22-s + 23-s − 3.87·24-s − 3.26·25-s + 9.86·26-s + 27-s − 3.12·28-s + ⋯
L(s)  = 1  − 1.67·2-s + 0.577·3-s + 1.81·4-s − 0.589·5-s − 0.968·6-s − 0.324·7-s − 1.37·8-s + 0.333·9-s + 0.989·10-s + 1.29·11-s + 1.04·12-s − 1.15·13-s + 0.545·14-s − 0.340·15-s + 0.483·16-s + 0.506·17-s − 0.559·18-s − 0.563·19-s − 1.07·20-s − 0.187·21-s − 2.18·22-s + 0.208·23-s − 0.791·24-s − 0.652·25-s + 1.93·26-s + 0.192·27-s − 0.590·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-1$
Analytic conductor: \(15.9780\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + 2.37T + 2T^{2} \)
5 \( 1 + 1.31T + 5T^{2} \)
7 \( 1 + 0.859T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 + 2.45T + 19T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + 3.17T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 - 8.47T + 59T^{2} \)
61 \( 1 + 6.39T + 61T^{2} \)
67 \( 1 - 4.74T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 8.54T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 0.795T + 89T^{2} \)
97 \( 1 - 3.92T + 97T^{2} \)
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

−8.906707116878849210428990540735, −8.097528670896476681481933623374, −7.38310019795180556257865664146, −6.96020473791284007231526869621, −5.94269005872411107390701131635, −4.47173938147046130968370335788, −3.54585864728428179566093507799, −2.42250102426845090066656055727, −1.40243800028650936821880551474, 0, 1.40243800028650936821880551474, 2.42250102426845090066656055727, 3.54585864728428179566093507799, 4.47173938147046130968370335788, 5.94269005872411107390701131635, 6.96020473791284007231526869621, 7.38310019795180556257865664146, 8.097528670896476681481933623374, 8.906707116878849210428990540735

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