Properties

Degree 2
Conductor $ 2 \cdot 2003 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.0106·3-s + 4-s − 1.19·5-s + 0.0106·6-s + 1.61·7-s − 8-s − 2.99·9-s + 1.19·10-s − 2.53·11-s − 0.0106·12-s + 6.61·13-s − 1.61·14-s + 0.0127·15-s + 16-s − 1.49·17-s + 2.99·18-s + 1.43·19-s − 1.19·20-s − 0.0172·21-s + 2.53·22-s − 0.350·23-s + 0.0106·24-s − 3.57·25-s − 6.61·26-s + 0.0639·27-s + 1.61·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.00615·3-s + 0.5·4-s − 0.534·5-s + 0.00435·6-s + 0.611·7-s − 0.353·8-s − 0.999·9-s + 0.377·10-s − 0.765·11-s − 0.00307·12-s + 1.83·13-s − 0.432·14-s + 0.00328·15-s + 0.250·16-s − 0.362·17-s + 0.707·18-s + 0.329·19-s − 0.267·20-s − 0.00376·21-s + 0.541·22-s − 0.0731·23-s + 0.00217·24-s − 0.714·25-s − 1.29·26-s + 0.0123·27-s + 0.305·28-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

\( d \)  =  \(2\)
\( N \)  =  \(4006\)    =    \(2 \cdot 2003\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4006} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4006,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;2003\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;2003\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
2003 \( 1 + T \)
good3 \( 1 + 0.0106T + 3T^{2} \)
5 \( 1 + 1.19T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
17 \( 1 + 1.49T + 17T^{2} \)
19 \( 1 - 1.43T + 19T^{2} \)
23 \( 1 + 0.350T + 23T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 + 0.851T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 + 5.21T + 53T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 7.90T + 67T^{2} \)
71 \( 1 + 7.53T + 71T^{2} \)
73 \( 1 + 9.66T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + 1.88T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 7.33T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.074010272904255054158011340707, −7.71943695886720528437234147183, −6.68532107250393106047714061367, −5.86762120696049676069455318273, −5.30548952576324801639477706705, −4.10978839593961841048375365949, −3.34826243811633108786078454189, −2.37018901333246815499644900356, −1.27241568428055522050979245185, 0, 1.27241568428055522050979245185, 2.37018901333246815499644900356, 3.34826243811633108786078454189, 4.10978839593961841048375365949, 5.30548952576324801639477706705, 5.86762120696049676069455318273, 6.68532107250393106047714061367, 7.71943695886720528437234147183, 8.074010272904255054158011340707

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