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 − 0.0866·3-s + 4-s + 2.47·5-s + 0.0866·6-s + 0.719·7-s − 8-s − 2.99·9-s − 2.47·10-s − 4.02·11-s − 0.0866·12-s − 1.17·13-s − 0.719·14-s − 0.214·15-s + 16-s + 0.882·17-s + 2.99·18-s + 0.0145·19-s + 2.47·20-s − 0.0623·21-s + 4.02·22-s − 5.06·23-s + 0.0866·24-s + 1.12·25-s + 1.17·26-s + 0.519·27-s + 0.719·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0500·3-s + 0.5·4-s + 1.10·5-s + 0.0353·6-s + 0.272·7-s − 0.353·8-s − 0.997·9-s − 0.782·10-s − 1.21·11-s − 0.0250·12-s − 0.325·13-s − 0.192·14-s − 0.0553·15-s + 0.250·16-s + 0.213·17-s + 0.705·18-s + 0.00333·19-s + 0.553·20-s − 0.0136·21-s + 0.857·22-s − 1.05·23-s + 0.0176·24-s + 0.225·25-s + 0.230·26-s + 0.0998·27-s + 0.136·28-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 \( 1 + 0.0866T + 3T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 - 0.719T + 7T^{2} \)
11 \( 1 + 4.02T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 - 0.882T + 17T^{2} \)
19 \( 1 - 0.0145T + 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 - 5.66T + 37T^{2} \)
41 \( 1 - 7.20T + 41T^{2} \)
43 \( 1 - 2.79T + 43T^{2} \)
47 \( 1 - 6.06T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 9.75T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 8.14T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 - 0.575T + 97T^{2} \)
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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

−7.988481287131898570224324498054, −7.73752003758449273761123675513, −6.42318457579535788134622999756, −5.99250787225271532676698946759, −5.30585832008917775433091251645, −4.42968283429212237487152078616, −2.82160765253929360204566925796, −2.56307848062865978667675259680, −1.40602757822128973514647459473, 0, 1.40602757822128973514647459473, 2.56307848062865978667675259680, 2.82160765253929360204566925796, 4.42968283429212237487152078616, 5.30585832008917775433091251645, 5.99250787225271532676698946759, 6.42318457579535788134622999756, 7.73752003758449273761123675513, 7.988481287131898570224324498054

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