Properties

Label 2-4006-1.1-c1-0-27
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.04·3-s + 4-s − 2.99·5-s − 1.04·6-s + 0.197·7-s − 8-s − 1.90·9-s + 2.99·10-s + 3.70·11-s + 1.04·12-s + 5.36·13-s − 0.197·14-s − 3.13·15-s + 16-s + 0.593·17-s + 1.90·18-s − 2.42·19-s − 2.99·20-s + 0.207·21-s − 3.70·22-s + 1.31·23-s − 1.04·24-s + 3.94·25-s − 5.36·26-s − 5.13·27-s + 0.197·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.604·3-s + 0.5·4-s − 1.33·5-s − 0.427·6-s + 0.0747·7-s − 0.353·8-s − 0.634·9-s + 0.945·10-s + 1.11·11-s + 0.302·12-s + 1.48·13-s − 0.0528·14-s − 0.808·15-s + 0.250·16-s + 0.143·17-s + 0.448·18-s − 0.557·19-s − 0.668·20-s + 0.0451·21-s − 0.789·22-s + 0.273·23-s − 0.213·24-s + 0.788·25-s − 1.05·26-s − 0.988·27-s + 0.0373·28-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: \(0\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.242324770\)
\(L(\frac12)\) \(\approx\) \(1.242324770\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.04T + 3T^{2} \)
5 \( 1 + 2.99T + 5T^{2} \)
7 \( 1 - 0.197T + 7T^{2} \)
11 \( 1 - 3.70T + 11T^{2} \)
13 \( 1 - 5.36T + 13T^{2} \)
17 \( 1 - 0.593T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 - 1.31T + 23T^{2} \)
29 \( 1 - 1.52T + 29T^{2} \)
31 \( 1 + 8.88T + 31T^{2} \)
37 \( 1 - 2.20T + 37T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 - 3.09T + 43T^{2} \)
47 \( 1 - 5.28T + 47T^{2} \)
53 \( 1 - 9.41T + 53T^{2} \)
59 \( 1 + 8.67T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 + 6.11T + 67T^{2} \)
71 \( 1 - 6.22T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 2.35T + 83T^{2} \)
89 \( 1 + 0.557T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−8.580535436724197635347461706131, −7.84714229017015100878650354633, −7.29325762232533480842028134002, −6.37345825946327206446096396352, −5.72081080761328664907893446814, −4.30460859659446789991969782071, −3.71759520804400844792305354049, −3.08610769903557919410496489841, −1.83570140827459685201203142864, −0.69729094257629431869293034867, 0.69729094257629431869293034867, 1.83570140827459685201203142864, 3.08610769903557919410496489841, 3.71759520804400844792305354049, 4.30460859659446789991969782071, 5.72081080761328664907893446814, 6.37345825946327206446096396352, 7.29325762232533480842028134002, 7.84714229017015100878650354633, 8.580535436724197635347461706131

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