Properties

Label 2-2001-1.1-c1-0-91
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.44·2-s + 3-s + 0.0738·4-s + 0.734·5-s − 1.44·6-s + 3.73·7-s + 2.77·8-s + 9-s − 1.05·10-s − 2.70·11-s + 0.0738·12-s − 2.87·13-s − 5.38·14-s + 0.734·15-s − 4.14·16-s − 2.75·17-s − 1.44·18-s − 5.54·19-s + 0.0542·20-s + 3.73·21-s + 3.89·22-s + 23-s + 2.77·24-s − 4.46·25-s + 4.13·26-s + 27-s + 0.275·28-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.577·3-s + 0.0369·4-s + 0.328·5-s − 0.587·6-s + 1.41·7-s + 0.980·8-s + 0.333·9-s − 0.334·10-s − 0.814·11-s + 0.0213·12-s − 0.796·13-s − 1.43·14-s + 0.189·15-s − 1.03·16-s − 0.667·17-s − 0.339·18-s − 1.27·19-s + 0.0121·20-s + 0.815·21-s + 0.829·22-s + 0.208·23-s + 0.566·24-s − 0.892·25-s + 0.811·26-s + 0.192·27-s + 0.0521·28-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 \( 1 + 1.44T + 2T^{2} \)
5 \( 1 - 0.734T + 5T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 + 2.70T + 11T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 1.81T + 41T^{2} \)
43 \( 1 - 0.181T + 43T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 + 0.662T + 53T^{2} \)
59 \( 1 + 0.174T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 0.990T + 73T^{2} \)
79 \( 1 + 2.96T + 79T^{2} \)
83 \( 1 - 1.08T + 83T^{2} \)
89 \( 1 - 5.21T + 89T^{2} \)
97 \( 1 + 3.97T + 97T^{2} \)
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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

−8.717118578015768752766380862944, −8.131549011821224595789246283597, −7.57710162090899735183185396189, −6.79547412556635043790966842538, −5.31282909916464794060561828894, −4.80422054537689023147445070983, −3.81429446373271666524148456148, −2.15497642808473395639846435875, −1.80691191805346908965086357817, 0, 1.80691191805346908965086357817, 2.15497642808473395639846435875, 3.81429446373271666524148456148, 4.80422054537689023147445070983, 5.31282909916464794060561828894, 6.79547412556635043790966842538, 7.57710162090899735183185396189, 8.131549011821224595789246283597, 8.717118578015768752766380862944

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