Properties

Label 2-4006-1.1-c1-0-99
Degree $2$
Conductor $4006$
Sign $-1$
Analytic cond. $31.9880$
Root an. cond. $5.65579$
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  + 2-s − 2.69·3-s + 4-s + 2.56·5-s − 2.69·6-s − 4.70·7-s + 8-s + 4.28·9-s + 2.56·10-s − 3.35·11-s − 2.69·12-s − 1.77·13-s − 4.70·14-s − 6.91·15-s + 16-s + 7.86·17-s + 4.28·18-s + 2.18·19-s + 2.56·20-s + 12.6·21-s − 3.35·22-s + 1.70·23-s − 2.69·24-s + 1.56·25-s − 1.77·26-s − 3.48·27-s − 4.70·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.55·3-s + 0.5·4-s + 1.14·5-s − 1.10·6-s − 1.77·7-s + 0.353·8-s + 1.42·9-s + 0.810·10-s − 1.01·11-s − 0.779·12-s − 0.493·13-s − 1.25·14-s − 1.78·15-s + 0.250·16-s + 1.90·17-s + 1.01·18-s + 0.500·19-s + 0.572·20-s + 2.77·21-s − 0.716·22-s + 0.355·23-s − 0.551·24-s + 0.312·25-s − 0.348·26-s − 0.669·27-s − 0.888·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4006\)    =    \(2 \cdot 2003\)
Sign: $-1$
Analytic conductor: \(31.9880\)
Root analytic conductor: \(5.65579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
2003 \( 1 - T \)
good3 \( 1 + 2.69T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 4.70T + 7T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 + 1.77T + 13T^{2} \)
17 \( 1 - 7.86T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 - 1.70T + 23T^{2} \)
29 \( 1 + 1.59T + 29T^{2} \)
31 \( 1 - 0.397T + 31T^{2} \)
37 \( 1 - 0.252T + 37T^{2} \)
41 \( 1 + 6.51T + 41T^{2} \)
43 \( 1 - 9.23T + 43T^{2} \)
47 \( 1 + 0.749T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + 2.84T + 59T^{2} \)
61 \( 1 + 0.902T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 3.25T + 97T^{2} \)
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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

−7.60833765218548653529072334047, −7.08319566422364502121551856094, −6.20739186747239175206100782319, −5.74253959680492938837338044244, −5.48130179308619944609286042164, −4.58023853255658606088243447389, −3.30856858799888617492304217351, −2.72271617681952282067758501198, −1.30572113957961668006216252844, 0, 1.30572113957961668006216252844, 2.72271617681952282067758501198, 3.30856858799888617492304217351, 4.58023853255658606088243447389, 5.48130179308619944609286042164, 5.74253959680492938837338044244, 6.20739186747239175206100782319, 7.08319566422364502121551856094, 7.60833765218548653529072334047

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