Properties

Label 2-4006-1.1-c1-0-11
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 − 0.857·3-s + 4-s − 2.26·5-s − 0.857·6-s − 3.52·7-s + 8-s − 2.26·9-s − 2.26·10-s − 1.16·11-s − 0.857·12-s − 5.73·13-s − 3.52·14-s + 1.94·15-s + 16-s + 6.08·17-s − 2.26·18-s + 4.64·19-s − 2.26·20-s + 3.02·21-s − 1.16·22-s − 4.86·23-s − 0.857·24-s + 0.151·25-s − 5.73·26-s + 4.51·27-s − 3.52·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.495·3-s + 0.5·4-s − 1.01·5-s − 0.350·6-s − 1.33·7-s + 0.353·8-s − 0.754·9-s − 0.717·10-s − 0.351·11-s − 0.247·12-s − 1.59·13-s − 0.941·14-s + 0.502·15-s + 0.250·16-s + 1.47·17-s − 0.533·18-s + 1.06·19-s − 0.507·20-s + 0.659·21-s − 0.248·22-s − 1.01·23-s − 0.175·24-s + 0.0302·25-s − 1.12·26-s + 0.869·27-s − 0.665·28-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

\(L(1)\) \(\approx\) \(0.7539299907\)
\(L(\frac12)\) \(\approx\) \(0.7539299907\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.857T + 3T^{2} \)
5 \( 1 + 2.26T + 5T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 - 6.08T + 17T^{2} \)
19 \( 1 - 4.64T + 19T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 + 4.30T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 4.01T + 37T^{2} \)
41 \( 1 + 0.986T + 41T^{2} \)
43 \( 1 + 1.40T + 43T^{2} \)
47 \( 1 - 2.45T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 + 4.65T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 - 8.57T + 73T^{2} \)
79 \( 1 - 9.05T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + 2.29T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−8.106984527166072652142562242724, −7.55119322051448668760587195901, −7.04613437160279802774877970726, −5.99051380676174076673301272877, −5.51739958893504655294286323087, −4.80262496287692892018091348005, −3.56368286049202513112333738562, −3.35042453253719927521707732022, −2.28822002018287489184924135860, −0.42547603994964618186768903357, 0.42547603994964618186768903357, 2.28822002018287489184924135860, 3.35042453253719927521707732022, 3.56368286049202513112333738562, 4.80262496287692892018091348005, 5.51739958893504655294286323087, 5.99051380676174076673301272877, 7.04613437160279802774877970726, 7.55119322051448668760587195901, 8.106984527166072652142562242724

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