Properties

Label 2-4006-1.1-c1-0-73
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 − 1.53·3-s + 4-s + 3.32·5-s − 1.53·6-s + 2.69·7-s + 8-s − 0.643·9-s + 3.32·10-s + 2.69·11-s − 1.53·12-s − 1.32·13-s + 2.69·14-s − 5.10·15-s + 16-s + 6.00·17-s − 0.643·18-s − 7.32·19-s + 3.32·20-s − 4.13·21-s + 2.69·22-s + 2.96·23-s − 1.53·24-s + 6.06·25-s − 1.32·26-s + 5.59·27-s + 2.69·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.886·3-s + 0.5·4-s + 1.48·5-s − 0.626·6-s + 1.01·7-s + 0.353·8-s − 0.214·9-s + 1.05·10-s + 0.811·11-s − 0.443·12-s − 0.367·13-s + 0.719·14-s − 1.31·15-s + 0.250·16-s + 1.45·17-s − 0.151·18-s − 1.67·19-s + 0.743·20-s − 0.901·21-s + 0.574·22-s + 0.618·23-s − 0.313·24-s + 1.21·25-s − 0.259·26-s + 1.07·27-s + 0.508·28-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

\(L(1)\) \(\approx\) \(3.536335947\)
\(L(\frac12)\) \(\approx\) \(3.536335947\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 3T^{2} \)
5 \( 1 - 3.32T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 + 7.32T + 19T^{2} \)
23 \( 1 - 2.96T + 23T^{2} \)
29 \( 1 + 3.76T + 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 - 8.07T + 37T^{2} \)
41 \( 1 + 0.334T + 41T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 - 9.76T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 - 2.79T + 59T^{2} \)
61 \( 1 + 6.21T + 61T^{2} \)
67 \( 1 - 6.21T + 67T^{2} \)
71 \( 1 - 7.65T + 71T^{2} \)
73 \( 1 - 8.06T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 1.24T + 97T^{2} \)
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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

−8.420154713603022580330796283422, −7.56718655727300745952845951465, −6.52528618135736018849617199806, −6.12548839872250960372467365938, −5.47214429611435661232571530283, −4.93214524800433457396780820685, −4.12656925089284701259731266427, −2.84241777380563779412603103207, −1.95407686487255827137623738600, −1.10322382335750770210550958897, 1.10322382335750770210550958897, 1.95407686487255827137623738600, 2.84241777380563779412603103207, 4.12656925089284701259731266427, 4.93214524800433457396780820685, 5.47214429611435661232571530283, 6.12548839872250960372467365938, 6.52528618135736018849617199806, 7.56718655727300745952845951465, 8.420154713603022580330796283422

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