Properties

Label 2-4002-1.1-c1-0-42
Degree $2$
Conductor $4002$
Sign $-1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 4.40·5-s − 6-s − 0.551·7-s − 8-s + 9-s + 4.40·10-s + 1.47·11-s + 12-s − 2.62·13-s + 0.551·14-s − 4.40·15-s + 16-s − 1.48·17-s − 18-s + 2.46·19-s − 4.40·20-s − 0.551·21-s − 1.47·22-s − 23-s − 24-s + 14.4·25-s + 2.62·26-s + 27-s − 0.551·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s − 0.208·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s + 0.444·11-s + 0.288·12-s − 0.727·13-s + 0.147·14-s − 1.13·15-s + 0.250·16-s − 0.360·17-s − 0.235·18-s + 0.566·19-s − 0.986·20-s − 0.120·21-s − 0.314·22-s − 0.208·23-s − 0.204·24-s + 2.88·25-s + 0.514·26-s + 0.192·27-s − 0.104·28-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.40T + 5T^{2} \)
7 \( 1 + 0.551T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 + 1.48T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
31 \( 1 - 5.87T + 31T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 + 0.176T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 - 9.14T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 6.61T + 73T^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 + 6.00T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.033425280244344871730167380733, −7.53429302683910003198798419067, −7.03064216557184640527348051845, −6.15614789561547454589583403927, −4.73397804536037602947924394714, −4.20492927554533029888724695857, −3.27663752488207807709485966493, −2.65462199985882722696557368439, −1.15136100559167426286202914305, 0, 1.15136100559167426286202914305, 2.65462199985882722696557368439, 3.27663752488207807709485966493, 4.20492927554533029888724695857, 4.73397804536037602947924394714, 6.15614789561547454589583403927, 7.03064216557184640527348051845, 7.53429302683910003198798419067, 8.033425280244344871730167380733

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