Properties

Label 2-2001-1.1-c1-0-99
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  + 1.17·2-s + 3-s − 0.629·4-s + 0.418·5-s + 1.17·6-s + 1.98·7-s − 3.07·8-s + 9-s + 0.490·10-s − 5.41·11-s − 0.629·12-s − 1.43·13-s + 2.32·14-s + 0.418·15-s − 2.34·16-s − 6.03·17-s + 1.17·18-s − 6.20·19-s − 0.263·20-s + 1.98·21-s − 6.33·22-s − 23-s − 3.07·24-s − 4.82·25-s − 1.68·26-s + 27-s − 1.24·28-s + ⋯
L(s)  = 1  + 0.827·2-s + 0.577·3-s − 0.314·4-s + 0.187·5-s + 0.478·6-s + 0.748·7-s − 1.08·8-s + 0.333·9-s + 0.155·10-s − 1.63·11-s − 0.181·12-s − 0.399·13-s + 0.620·14-s + 0.108·15-s − 0.586·16-s − 1.46·17-s + 0.275·18-s − 1.42·19-s − 0.0589·20-s + 0.432·21-s − 1.35·22-s − 0.208·23-s − 0.628·24-s − 0.964·25-s − 0.330·26-s + 0.192·27-s − 0.235·28-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: Trivial
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 \( 1 - 1.17T + 2T^{2} \)
5 \( 1 - 0.418T + 5T^{2} \)
7 \( 1 - 1.98T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
17 \( 1 + 6.03T + 17T^{2} \)
19 \( 1 + 6.20T + 19T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 8.27T + 41T^{2} \)
43 \( 1 - 6.28T + 43T^{2} \)
47 \( 1 - 6.99T + 47T^{2} \)
53 \( 1 - 5.10T + 53T^{2} \)
59 \( 1 - 9.51T + 59T^{2} \)
61 \( 1 + 4.79T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 7.30T + 71T^{2} \)
73 \( 1 + 4.61T + 73T^{2} \)
79 \( 1 + 4.07T + 79T^{2} \)
83 \( 1 - 3.71T + 83T^{2} \)
89 \( 1 + 0.547T + 89T^{2} \)
97 \( 1 + 8.58T + 97T^{2} \)
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

−8.752116622708251397889983853715, −8.026553781665255769515691803386, −7.31542004883182494986005000755, −6.11610945180267226025088199318, −5.43133218433426038531897130407, −4.48977406172041110571684228508, −4.10899390878836019766944822112, −2.67389853080316596280155605773, −2.16668558272558830836989764920, 0, 2.16668558272558830836989764920, 2.67389853080316596280155605773, 4.10899390878836019766944822112, 4.48977406172041110571684228508, 5.43133218433426038531897130407, 6.11610945180267226025088199318, 7.31542004883182494986005000755, 8.026553781665255769515691803386, 8.752116622708251397889983853715

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