Properties

Label 2-4002-1.1-c1-0-34
Degree $2$
Conductor $4002$
Sign $1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 12-s − 2·13-s + 4·15-s + 16-s + 4·17-s − 18-s + 4·19-s + 4·20-s − 23-s − 24-s + 11·25-s + 2·26-s + 27-s − 29-s − 4·30-s − 32-s − 4·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s − 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.208·23-s − 0.204·24-s + 11/5·25-s + 0.392·26-s + 0.192·27-s − 0.185·29-s − 0.730·30-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(31.9561\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.629011397\)
\(L(\frac12)\) \(\approx\) \(2.629011397\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.551231241777373500582267250073, −7.81813125460748135589814895645, −7.05856558855263757560069044572, −6.33172175295868445703252356619, −5.54733361459313485443848642347, −4.95494042039172426130458969123, −3.50989093785209109916887500252, −2.66735401582025278323523094777, −1.93647848526306893950400856932, −1.07660477402409874239789605676, 1.07660477402409874239789605676, 1.93647848526306893950400856932, 2.66735401582025278323523094777, 3.50989093785209109916887500252, 4.95494042039172426130458969123, 5.54733361459313485443848642347, 6.33172175295868445703252356619, 7.05856558855263757560069044572, 7.81813125460748135589814895645, 8.551231241777373500582267250073

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