Properties

Label 2-4006-1.1-c1-0-132
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 + 2.43·3-s + 4-s + 2.88·5-s + 2.43·6-s + 0.287·7-s + 8-s + 2.94·9-s + 2.88·10-s + 5.71·11-s + 2.43·12-s − 6.12·13-s + 0.287·14-s + 7.02·15-s + 16-s + 5.27·17-s + 2.94·18-s + 1.13·19-s + 2.88·20-s + 0.700·21-s + 5.71·22-s + 3.27·23-s + 2.43·24-s + 3.30·25-s − 6.12·26-s − 0.122·27-s + 0.287·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.40·3-s + 0.5·4-s + 1.28·5-s + 0.995·6-s + 0.108·7-s + 0.353·8-s + 0.983·9-s + 0.911·10-s + 1.72·11-s + 0.704·12-s − 1.69·13-s + 0.0767·14-s + 1.81·15-s + 0.250·16-s + 1.27·17-s + 0.695·18-s + 0.260·19-s + 0.644·20-s + 0.152·21-s + 1.21·22-s + 0.682·23-s + 0.497·24-s + 0.660·25-s − 1.20·26-s − 0.0236·27-s + 0.0542·28-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

\(L(1)\) \(\approx\) \(6.777081524\)
\(L(\frac12)\) \(\approx\) \(6.777081524\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 2.88T + 5T^{2} \)
7 \( 1 - 0.287T + 7T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + 6.12T + 13T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
19 \( 1 - 1.13T + 19T^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + 5.66T + 29T^{2} \)
31 \( 1 + 4.48T + 31T^{2} \)
37 \( 1 - 3.87T + 37T^{2} \)
41 \( 1 + 7.30T + 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 0.910T + 53T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 - 1.33T + 61T^{2} \)
67 \( 1 + 5.48T + 67T^{2} \)
71 \( 1 + 6.21T + 71T^{2} \)
73 \( 1 + 8.01T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 - 1.05T + 83T^{2} \)
89 \( 1 + 0.512T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−8.525227670337103638274830083872, −7.59313161821298093080696371946, −7.02672698310075445823588067576, −6.24117358194903326913212913041, −5.35283926296658437240129245690, −4.70227018648306005584458387357, −3.51977850080257298208827300160, −3.12971434562439036442090082646, −1.99332803755348228446941199629, −1.58842122894774098818267682362, 1.58842122894774098818267682362, 1.99332803755348228446941199629, 3.12971434562439036442090082646, 3.51977850080257298208827300160, 4.70227018648306005584458387357, 5.35283926296658437240129245690, 6.24117358194903326913212913041, 7.02672698310075445823588067576, 7.59313161821298093080696371946, 8.525227670337103638274830083872

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