Properties

Label 2-2001-1.1-c1-0-100
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  + 1.21·2-s + 3-s − 0.534·4-s + 3.03·5-s + 1.21·6-s − 3.40·7-s − 3.06·8-s + 9-s + 3.67·10-s − 3.85·11-s − 0.534·12-s − 5.15·13-s − 4.12·14-s + 3.03·15-s − 2.64·16-s − 2.07·17-s + 1.21·18-s − 0.0953·19-s − 1.62·20-s − 3.40·21-s − 4.66·22-s + 23-s − 3.06·24-s + 4.22·25-s − 6.24·26-s + 27-s + 1.82·28-s + ⋯
L(s)  = 1  + 0.856·2-s + 0.577·3-s − 0.267·4-s + 1.35·5-s + 0.494·6-s − 1.28·7-s − 1.08·8-s + 0.333·9-s + 1.16·10-s − 1.16·11-s − 0.154·12-s − 1.43·13-s − 1.10·14-s + 0.784·15-s − 0.661·16-s − 0.503·17-s + 0.285·18-s − 0.0218·19-s − 0.362·20-s − 0.743·21-s − 0.995·22-s + 0.208·23-s − 0.626·24-s + 0.845·25-s − 1.22·26-s + 0.192·27-s + 0.344·28-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 \( 1 - 1.21T + 2T^{2} \)
5 \( 1 - 3.03T + 5T^{2} \)
7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + 5.15T + 13T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + 0.0953T + 19T^{2} \)
31 \( 1 + 3.70T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 4.31T + 41T^{2} \)
43 \( 1 - 2.81T + 43T^{2} \)
47 \( 1 + 2.48T + 47T^{2} \)
53 \( 1 + 6.40T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 5.67T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 0.679T + 79T^{2} \)
83 \( 1 - 4.99T + 83T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−9.148072033930318349995899269289, −7.987431199710081814329568644024, −6.95531066663036131675161076552, −6.26375253028128632612957427214, −5.37766231659673463634664555311, −4.89581595556352822188163168416, −3.66120216551366245954769805317, −2.78520282663666424411302311339, −2.21442877862716523103219016889, 0, 2.21442877862716523103219016889, 2.78520282663666424411302311339, 3.66120216551366245954769805317, 4.89581595556352822188163168416, 5.37766231659673463634664555311, 6.26375253028128632612957427214, 6.95531066663036131675161076552, 7.987431199710081814329568644024, 9.148072033930318349995899269289

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