Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.255·3-s + 4-s + 2.80·5-s − 0.255·6-s + 3.25·7-s − 8-s − 2.93·9-s − 2.80·10-s − 2.27·11-s + 0.255·12-s + 0.713·13-s − 3.25·14-s + 0.715·15-s + 16-s − 3.04·17-s + 2.93·18-s − 1.81·19-s + 2.80·20-s + 0.832·21-s + 2.27·22-s − 6.95·23-s − 0.255·24-s + 2.84·25-s − 0.713·26-s − 1.51·27-s + 3.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.147·3-s + 0.5·4-s + 1.25·5-s − 0.104·6-s + 1.23·7-s − 0.353·8-s − 0.978·9-s − 0.885·10-s − 0.686·11-s + 0.0737·12-s + 0.197·13-s − 0.870·14-s + 0.184·15-s + 0.250·16-s − 0.739·17-s + 0.691·18-s − 0.416·19-s + 0.626·20-s + 0.181·21-s + 0.485·22-s − 1.45·23-s − 0.0521·24-s + 0.569·25-s − 0.139·26-s − 0.291·27-s + 0.615·28-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 \( 1 - 0.255T + 3T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
13 \( 1 - 0.713T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
19 \( 1 + 1.81T + 19T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
31 \( 1 - 0.0671T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 6.82T + 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 + 0.951T + 61T^{2} \)
67 \( 1 + 8.87T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 2.22T + 73T^{2} \)
79 \( 1 + 8.86T + 79T^{2} \)
83 \( 1 + 3.93T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 + 2.72T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.221673515730364271572294581430, −7.63314645391324759079831352682, −6.55629805778886879557148454351, −5.84766387511491024636927014140, −5.34224000216334438804102750245, −4.38127977818799416380068570763, −3.07808109769220549947260971087, −2.04992138562011833552796348906, −1.76094289670544973436736015896, 0, 1.76094289670544973436736015896, 2.04992138562011833552796348906, 3.07808109769220549947260971087, 4.38127977818799416380068570763, 5.34224000216334438804102750245, 5.84766387511491024636927014140, 6.55629805778886879557148454351, 7.63314645391324759079831352682, 8.221673515730364271572294581430

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