Properties

Label 2-2001-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.676·2-s − 3-s − 1.54·4-s − 1.80·5-s − 0.676·6-s − 4.44·7-s − 2.39·8-s + 9-s − 1.22·10-s − 1.20·11-s + 1.54·12-s − 3.05·13-s − 3.00·14-s + 1.80·15-s + 1.46·16-s − 6.57·17-s + 0.676·18-s − 2.24·19-s + 2.78·20-s + 4.44·21-s − 0.814·22-s − 23-s + 2.39·24-s − 1.73·25-s − 2.06·26-s − 27-s + 6.85·28-s + ⋯
L(s)  = 1  + 0.478·2-s − 0.577·3-s − 0.771·4-s − 0.807·5-s − 0.276·6-s − 1.68·7-s − 0.847·8-s + 0.333·9-s − 0.386·10-s − 0.362·11-s + 0.445·12-s − 0.847·13-s − 0.803·14-s + 0.466·15-s + 0.365·16-s − 1.59·17-s + 0.159·18-s − 0.514·19-s + 0.622·20-s + 0.970·21-s − 0.173·22-s − 0.208·23-s + 0.489·24-s − 0.347·25-s − 0.405·26-s − 0.192·27-s + 1.29·28-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: \(0\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1302117296\)
\(L(\frac12)\) \(\approx\) \(0.1302117296\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.676T + 2T^{2} \)
5 \( 1 + 1.80T + 5T^{2} \)
7 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + 3.05T + 13T^{2} \)
17 \( 1 + 6.57T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 + 1.57T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 8.54T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 6.15T + 59T^{2} \)
61 \( 1 + 0.352T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 + 8.06T + 73T^{2} \)
79 \( 1 - 4.13T + 79T^{2} \)
83 \( 1 + 1.05T + 83T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 - 8.49T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.169341538801868643250707660369, −8.517232568036867919317184407595, −7.41420399387724238910764075184, −6.64260533106541082858677551156, −6.00092279301179236486645061691, −4.99797255075645579596104146269, −4.25234591866386174172466179503, −3.55279531965561941314505334824, −2.53203189230540012192199565552, −0.21398763684061749363130051731, 0.21398763684061749363130051731, 2.53203189230540012192199565552, 3.55279531965561941314505334824, 4.25234591866386174172466179503, 4.99797255075645579596104146269, 6.00092279301179236486645061691, 6.64260533106541082858677551156, 7.41420399387724238910764075184, 8.517232568036867919317184407595, 9.169341538801868643250707660369

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