Properties

Label 2-4002-1.1-c1-0-3
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 + 1.46·5-s + 6-s − 2.29·7-s − 8-s + 9-s − 1.46·10-s − 1.24·11-s − 12-s − 6.90·13-s + 2.29·14-s − 1.46·15-s + 16-s + 5.12·17-s − 18-s − 3.85·19-s + 1.46·20-s + 2.29·21-s + 1.24·22-s + 23-s + 24-s − 2.85·25-s + 6.90·26-s − 27-s − 2.29·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.655·5-s + 0.408·6-s − 0.867·7-s − 0.353·8-s + 0.333·9-s − 0.463·10-s − 0.375·11-s − 0.288·12-s − 1.91·13-s + 0.613·14-s − 0.378·15-s + 0.250·16-s + 1.24·17-s − 0.235·18-s − 0.883·19-s + 0.327·20-s + 0.500·21-s + 0.265·22-s + 0.208·23-s + 0.204·24-s − 0.570·25-s + 1.35·26-s − 0.192·27-s − 0.433·28-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

\(L(1)\) \(\approx\) \(0.6657857858\)
\(L(\frac12)\) \(\approx\) \(0.6657857858\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.46T + 5T^{2} \)
7 \( 1 + 2.29T + 7T^{2} \)
11 \( 1 + 1.24T + 11T^{2} \)
13 \( 1 + 6.90T + 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 3.85T + 19T^{2} \)
31 \( 1 - 3.24T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 3.53T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 - 6.29T + 47T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + 7.36T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 6.17T + 79T^{2} \)
83 \( 1 - 1.61T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 2.68T + 97T^{2} \)
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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

−8.467614755818057075326147825561, −7.62762383713886480494401875781, −6.99913235269792924328859452546, −6.36207747869269921871627245759, −5.51874732136206612015158149986, −5.00066059664194345080164445834, −3.73190445303366159665508059577, −2.68426144999371300567791138423, −1.94125957294907385844774291203, −0.51000000635580593785659963192, 0.51000000635580593785659963192, 1.94125957294907385844774291203, 2.68426144999371300567791138423, 3.73190445303366159665508059577, 5.00066059664194345080164445834, 5.51874732136206612015158149986, 6.36207747869269921871627245759, 6.99913235269792924328859452546, 7.62762383713886480494401875781, 8.467614755818057075326147825561

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