Properties

Label 2-4002-1.1-c1-0-28
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 1.65·5-s + 6-s − 1.05·7-s + 8-s + 9-s − 1.65·10-s − 2.77·11-s + 12-s + 5.36·13-s − 1.05·14-s − 1.65·15-s + 16-s + 5.11·17-s + 18-s − 0.230·19-s − 1.65·20-s − 1.05·21-s − 2.77·22-s + 23-s + 24-s − 2.24·25-s + 5.36·26-s + 27-s − 1.05·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.741·5-s + 0.408·6-s − 0.399·7-s + 0.353·8-s + 0.333·9-s − 0.524·10-s − 0.836·11-s + 0.288·12-s + 1.48·13-s − 0.282·14-s − 0.428·15-s + 0.250·16-s + 1.24·17-s + 0.235·18-s − 0.0528·19-s − 0.370·20-s − 0.230·21-s − 0.591·22-s + 0.208·23-s + 0.204·24-s − 0.449·25-s + 1.05·26-s + 0.192·27-s − 0.199·28-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

\(L(1)\) \(\approx\) \(3.332686705\)
\(L(\frac12)\) \(\approx\) \(3.332686705\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 + 1.05T + 7T^{2} \)
11 \( 1 + 2.77T + 11T^{2} \)
13 \( 1 - 5.36T + 13T^{2} \)
17 \( 1 - 5.11T + 17T^{2} \)
19 \( 1 + 0.230T + 19T^{2} \)
31 \( 1 - 3.97T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 - 5.80T + 41T^{2} \)
43 \( 1 - 2.66T + 43T^{2} \)
47 \( 1 - 4.87T + 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 - 6.68T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + 7.07T + 67T^{2} \)
71 \( 1 - 3.51T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 2.10T + 79T^{2} \)
83 \( 1 + 1.00T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 0.785T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.233050067656247049022802401632, −7.77588549798837505758050066715, −7.04870718222913705343404270088, −6.11847650518150485308171142123, −5.50250020839375996009761349117, −4.50934684592448873533060077993, −3.63254390445407893727521759189, −3.30046838257469907063367438776, −2.24591889743358404158730500538, −0.938313745523045900913261984243, 0.938313745523045900913261984243, 2.24591889743358404158730500538, 3.30046838257469907063367438776, 3.63254390445407893727521759189, 4.50934684592448873533060077993, 5.50250020839375996009761349117, 6.11847650518150485308171142123, 7.04870718222913705343404270088, 7.77588549798837505758050066715, 8.233050067656247049022802401632

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