Properties

Label 2-4002-1.1-c1-0-49
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 + 4.15·5-s − 6-s + 0.461·7-s + 8-s + 9-s + 4.15·10-s + 2.29·11-s − 12-s − 2.16·13-s + 0.461·14-s − 4.15·15-s + 16-s + 3.68·17-s + 18-s + 0.461·19-s + 4.15·20-s − 0.461·21-s + 2.29·22-s + 23-s − 24-s + 12.2·25-s − 2.16·26-s − 27-s + 0.461·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.85·5-s − 0.408·6-s + 0.174·7-s + 0.353·8-s + 0.333·9-s + 1.31·10-s + 0.691·11-s − 0.288·12-s − 0.600·13-s + 0.123·14-s − 1.07·15-s + 0.250·16-s + 0.894·17-s + 0.235·18-s + 0.105·19-s + 0.929·20-s − 0.100·21-s + 0.488·22-s + 0.208·23-s − 0.204·24-s + 2.45·25-s − 0.424·26-s − 0.192·27-s + 0.0872·28-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

\(L(1)\) \(\approx\) \(4.037134488\)
\(L(\frac12)\) \(\approx\) \(4.037134488\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.15T + 5T^{2} \)
7 \( 1 - 0.461T + 7T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 + 2.16T + 13T^{2} \)
17 \( 1 - 3.68T + 17T^{2} \)
19 \( 1 - 0.461T + 19T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 + 7.35T + 37T^{2} \)
41 \( 1 - 1.98T + 41T^{2} \)
43 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 7.91T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 - 7.56T + 61T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 + 7.96T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 1.22T + 83T^{2} \)
89 \( 1 - 1.14T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.550472489428161534067156541147, −7.35747533725142878019463852775, −6.72816406049138503336846695626, −6.10730896872295301801781009240, −5.35152557775888084287255889485, −5.09107484140501474947857933616, −3.95391306704125882280871903823, −2.88268149853655992603567265294, −1.96223556634589965983488624934, −1.18551957697084490724709988211, 1.18551957697084490724709988211, 1.96223556634589965983488624934, 2.88268149853655992603567265294, 3.95391306704125882280871903823, 5.09107484140501474947857933616, 5.35152557775888084287255889485, 6.10730896872295301801781009240, 6.72816406049138503336846695626, 7.35747533725142878019463852775, 8.550472489428161534067156541147

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