Properties

Label 2-4006-1.1-c1-0-3
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 − 1.01·3-s + 4-s − 0.411·5-s + 1.01·6-s − 2.82·7-s − 8-s − 1.96·9-s + 0.411·10-s + 1.33·11-s − 1.01·12-s − 4.79·13-s + 2.82·14-s + 0.417·15-s + 16-s − 2.87·17-s + 1.96·18-s + 4.06·19-s − 0.411·20-s + 2.87·21-s − 1.33·22-s − 3.97·23-s + 1.01·24-s − 4.83·25-s + 4.79·26-s + 5.04·27-s − 2.82·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.586·3-s + 0.5·4-s − 0.183·5-s + 0.414·6-s − 1.06·7-s − 0.353·8-s − 0.656·9-s + 0.130·10-s + 0.401·11-s − 0.293·12-s − 1.33·13-s + 0.756·14-s + 0.107·15-s + 0.250·16-s − 0.698·17-s + 0.463·18-s + 0.932·19-s − 0.0919·20-s + 0.627·21-s − 0.283·22-s − 0.828·23-s + 0.207·24-s − 0.966·25-s + 0.940·26-s + 0.971·27-s − 0.534·28-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

\(L(1)\) \(\approx\) \(0.2267851516\)
\(L(\frac12)\) \(\approx\) \(0.2267851516\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.01T + 3T^{2} \)
5 \( 1 + 0.411T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 1.33T + 11T^{2} \)
13 \( 1 + 4.79T + 13T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 - 4.06T + 19T^{2} \)
23 \( 1 + 3.97T + 23T^{2} \)
29 \( 1 + 0.335T + 29T^{2} \)
31 \( 1 - 0.459T + 31T^{2} \)
37 \( 1 + 8.51T + 37T^{2} \)
41 \( 1 - 0.433T + 41T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 3.83T + 53T^{2} \)
59 \( 1 + 9.99T + 59T^{2} \)
61 \( 1 + 1.34T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 0.0392T + 71T^{2} \)
73 \( 1 - 5.14T + 73T^{2} \)
79 \( 1 - 6.86T + 79T^{2} \)
83 \( 1 + 5.20T + 83T^{2} \)
89 \( 1 + 6.43T + 89T^{2} \)
97 \( 1 - 2.23T + 97T^{2} \)
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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

−8.448440264952547638624244961981, −7.74803584270069498290162499319, −6.87078726569953199679447724235, −6.45432145876094012398766869012, −5.60358003016343736777323738048, −4.86081521277676482840749094738, −3.65875409040447037095354479459, −2.89754017500172830526982814286, −1.87196887447329131903425421670, −0.29408151696601778671310418384, 0.29408151696601778671310418384, 1.87196887447329131903425421670, 2.89754017500172830526982814286, 3.65875409040447037095354479459, 4.86081521277676482840749094738, 5.60358003016343736777323738048, 6.45432145876094012398766869012, 6.87078726569953199679447724235, 7.74803584270069498290162499319, 8.448440264952547638624244961981

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