Properties

Label 2-4006-1.1-c1-0-80
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 + 2.60·3-s + 4-s + 1.05·5-s − 2.60·6-s + 3.90·7-s − 8-s + 3.76·9-s − 1.05·10-s − 1.93·11-s + 2.60·12-s + 2.83·13-s − 3.90·14-s + 2.73·15-s + 16-s + 3.73·17-s − 3.76·18-s − 1.04·19-s + 1.05·20-s + 10.1·21-s + 1.93·22-s − 0.280·23-s − 2.60·24-s − 3.89·25-s − 2.83·26-s + 1.98·27-s + 3.90·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.469·5-s − 1.06·6-s + 1.47·7-s − 0.353·8-s + 1.25·9-s − 0.332·10-s − 0.584·11-s + 0.750·12-s + 0.787·13-s − 1.04·14-s + 0.705·15-s + 0.250·16-s + 0.906·17-s − 0.886·18-s − 0.240·19-s + 0.234·20-s + 2.21·21-s + 0.413·22-s − 0.0583·23-s − 0.530·24-s − 0.779·25-s − 0.556·26-s + 0.381·27-s + 0.738·28-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

\(L(1)\) \(\approx\) \(3.336806608\)
\(L(\frac12)\) \(\approx\) \(3.336806608\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 - 1.05T + 5T^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 + 1.93T + 11T^{2} \)
13 \( 1 - 2.83T + 13T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 + 1.04T + 19T^{2} \)
23 \( 1 + 0.280T + 23T^{2} \)
29 \( 1 - 9.47T + 29T^{2} \)
31 \( 1 + 0.699T + 31T^{2} \)
37 \( 1 + 3.36T + 37T^{2} \)
41 \( 1 - 7.58T + 41T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + 6.94T + 47T^{2} \)
53 \( 1 + 4.40T + 53T^{2} \)
59 \( 1 - 6.13T + 59T^{2} \)
61 \( 1 - 6.04T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + 1.02T + 71T^{2} \)
73 \( 1 - 0.806T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 5.03T + 83T^{2} \)
89 \( 1 - 6.52T + 89T^{2} \)
97 \( 1 - 9.56T + 97T^{2} \)
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

−8.572989999139362327207618885683, −7.85460947863122318690079375466, −7.53054167697114578774478513791, −6.38844121397986411219600768203, −5.49200655496347961051319995371, −4.59192970789156247740743340584, −3.59631332880332225953570559128, −2.71078924656190184127211258141, −1.94069632044529288563475710367, −1.21592926220285357129408492626, 1.21592926220285357129408492626, 1.94069632044529288563475710367, 2.71078924656190184127211258141, 3.59631332880332225953570559128, 4.59192970789156247740743340584, 5.49200655496347961051319995371, 6.38844121397986411219600768203, 7.53054167697114578774478513791, 7.85460947863122318690079375466, 8.572989999139362327207618885683

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