Properties

Label 2-4002-1.1-c1-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3.29·5-s + 6-s + 2.46·7-s − 8-s + 9-s + 3.29·10-s + 5.48·11-s − 12-s − 0.169·13-s − 2.46·14-s + 3.29·15-s + 16-s + 0.362·17-s − 18-s + 4.85·19-s − 3.29·20-s − 2.46·21-s − 5.48·22-s + 23-s + 24-s + 5.85·25-s + 0.169·26-s − 27-s + 2.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.47·5-s + 0.408·6-s + 0.931·7-s − 0.353·8-s + 0.333·9-s + 1.04·10-s + 1.65·11-s − 0.288·12-s − 0.0470·13-s − 0.658·14-s + 0.850·15-s + 0.250·16-s + 0.0879·17-s − 0.235·18-s + 1.11·19-s − 0.736·20-s − 0.538·21-s − 1.16·22-s + 0.208·23-s + 0.204·24-s + 1.17·25-s + 0.0332·26-s − 0.192·27-s + 0.465·28-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: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067834911\)
\(L(\frac12)\) \(\approx\) \(1.067834911\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.29T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 - 5.48T + 11T^{2} \)
13 \( 1 + 0.169T + 13T^{2} \)
17 \( 1 - 0.362T + 17T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
31 \( 1 + 3.48T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 7.95T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 1.53T + 47T^{2} \)
53 \( 1 + 5.21T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 9.27T + 61T^{2} \)
67 \( 1 + 6.34T + 67T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
79 \( 1 + 3.38T + 79T^{2} \)
83 \( 1 - 5.55T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−8.354326756415178472931518247419, −7.55586794163289647954787963463, −7.37753125595047426506835233865, −6.37670527755221752939393554408, −5.57847806407679861399926157271, −4.43610033059010684345837261443, −4.06031899064603669674149599849, −2.98834407728878027897381962259, −1.49845518981383636821181098975, −0.75341244899967336848972577795, 0.75341244899967336848972577795, 1.49845518981383636821181098975, 2.98834407728878027897381962259, 4.06031899064603669674149599849, 4.43610033059010684345837261443, 5.57847806407679861399926157271, 6.37670527755221752939393554408, 7.37753125595047426506835233865, 7.55586794163289647954787963463, 8.354326756415178472931518247419

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