Properties

Label 2-4002-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s − 3·7-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 3·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s − 3·21-s − 2·22-s + 23-s − 24-s + 4·25-s + 27-s − 3·28-s − 29-s + 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.566·28-s − 0.185·29-s + 0.547·30-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.181287385165343335154846329483, −7.51030725198775318183254953675, −6.77913959791946228009258145967, −6.25910920730510727427596292045, −4.97029829415090071042220572418, −3.88357020461952428072968382897, −3.43432195432228343033354965317, −2.60670009310683293711334821473, −1.18955940393403893990759813368, 0, 1.18955940393403893990759813368, 2.60670009310683293711334821473, 3.43432195432228343033354965317, 3.88357020461952428072968382897, 4.97029829415090071042220572418, 6.25910920730510727427596292045, 6.77913959791946228009258145967, 7.51030725198775318183254953675, 8.181287385165343335154846329483

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