Properties

Label 2-4006-1.1-c1-0-135
Degree $2$
Conductor $4006$
Sign $1$
Analytic cond. $31.9880$
Root an. cond. $5.65579$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.99·3-s + 4-s + 3.41·5-s + 1.99·6-s + 4.21·7-s + 8-s + 0.971·9-s + 3.41·10-s − 3.81·11-s + 1.99·12-s + 2.36·13-s + 4.21·14-s + 6.79·15-s + 16-s + 4.97·17-s + 0.971·18-s − 6.46·19-s + 3.41·20-s + 8.40·21-s − 3.81·22-s − 2.22·23-s + 1.99·24-s + 6.63·25-s + 2.36·26-s − 4.04·27-s + 4.21·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.5·4-s + 1.52·5-s + 0.813·6-s + 1.59·7-s + 0.353·8-s + 0.323·9-s + 1.07·10-s − 1.14·11-s + 0.575·12-s + 0.655·13-s + 1.12·14-s + 1.75·15-s + 0.250·16-s + 1.20·17-s + 0.229·18-s − 1.48·19-s + 0.762·20-s + 1.83·21-s − 0.812·22-s − 0.463·23-s + 0.406·24-s + 1.32·25-s + 0.463·26-s − 0.777·27-s + 0.796·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4006\)    =    \(2 \cdot 2003\)
Sign: $1$
Analytic conductor: \(31.9880\)
Root analytic conductor: \(5.65579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.765924202\)
\(L(\frac12)\) \(\approx\) \(6.765924202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2003 \( 1 + T \)
good3 \( 1 - 1.99T + 3T^{2} \)
5 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 - 4.21T + 7T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
13 \( 1 - 2.36T + 13T^{2} \)
17 \( 1 - 4.97T + 17T^{2} \)
19 \( 1 + 6.46T + 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 1.15T + 41T^{2} \)
43 \( 1 + 0.504T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 0.366T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 5.12T + 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 + 5.13T + 71T^{2} \)
73 \( 1 + 6.16T + 73T^{2} \)
79 \( 1 + 3.10T + 79T^{2} \)
83 \( 1 - 4.91T + 83T^{2} \)
89 \( 1 + 6.05T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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

−8.377305609108079802957671238057, −7.900234931407984976602003041547, −7.04698985862343469353794292551, −5.84040977368291669783003806189, −5.56782166202939644635478962271, −4.74682631538558091142916122891, −3.78748575743920643038300825793, −2.79760201233827928157364902869, −2.04478030465330577898286788933, −1.60754384117650729620789133006, 1.60754384117650729620789133006, 2.04478030465330577898286788933, 2.79760201233827928157364902869, 3.78748575743920643038300825793, 4.74682631538558091142916122891, 5.56782166202939644635478962271, 5.84040977368291669783003806189, 7.04698985862343469353794292551, 7.900234931407984976602003041547, 8.377305609108079802957671238057

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