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Properties

Label 2-2006-1.1-c1-0-38
Degree $2$
Conductor $2006$
Sign $-1$
Analytic cond. $16.0179$
Root an. cond. $4.00224$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 3·5-s − 3·6-s − 7-s + 8-s + 6·9-s − 3·10-s − 3·12-s + 4·13-s − 14-s + 9·15-s + 16-s + 17-s + 6·18-s − 19-s − 3·20-s + 3·21-s − 2·23-s − 3·24-s + 4·25-s + 4·26-s − 9·27-s − 28-s + 7·29-s + 9·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 0.948·10-s − 0.866·12-s + 1.10·13-s − 0.267·14-s + 2.32·15-s + 1/4·16-s + 0.242·17-s + 1.41·18-s − 0.229·19-s − 0.670·20-s + 0.654·21-s − 0.417·23-s − 0.612·24-s + 4/5·25-s + 0.784·26-s − 1.73·27-s − 0.188·28-s + 1.29·29-s + 1.64·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2006\)    =    \(2 \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(16.0179\)
Root analytic conductor: \(4.00224\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2006,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
17 \( 1 - T \)
59 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.583297532252097654946828110490, −7.71277856405851375483205907131, −6.91762119556739052353490713830, −6.24149909335142667872628732981, −5.63861615120294364961505680254, −4.60852520211246656031366640065, −4.10108950464063525749218434810, −3.16895257989102700484753773685, −1.28389304238819033579934784144, 0, 1.28389304238819033579934784144, 3.16895257989102700484753773685, 4.10108950464063525749218434810, 4.60852520211246656031366640065, 5.63861615120294364961505680254, 6.24149909335142667872628732981, 6.91762119556739052353490713830, 7.71277856405851375483205907131, 8.583297532252097654946828110490

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