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 − 3.11·3-s + 4-s + 1.57·5-s + 3.11·6-s + 2.57·7-s − 8-s + 6.72·9-s − 1.57·10-s − 2.27·11-s − 3.11·12-s + 0.272·13-s − 2.57·14-s − 4.90·15-s + 16-s + 3.49·17-s − 6.72·18-s + 3.02·19-s + 1.57·20-s − 8.03·21-s + 2.27·22-s + 1.09·23-s + 3.11·24-s − 2.53·25-s − 0.272·26-s − 11.6·27-s + 2.57·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.702·5-s + 1.27·6-s + 0.973·7-s − 0.353·8-s + 2.24·9-s − 0.496·10-s − 0.686·11-s − 0.900·12-s + 0.0755·13-s − 0.688·14-s − 1.26·15-s + 0.250·16-s + 0.847·17-s − 1.58·18-s + 0.694·19-s + 0.351·20-s − 1.75·21-s + 0.485·22-s + 0.229·23-s + 0.636·24-s − 0.506·25-s − 0.0533·26-s − 2.23·27-s + 0.486·28-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 \( 1 + 3.11T + 3T^{2} \)
5 \( 1 - 1.57T + 5T^{2} \)
7 \( 1 - 2.57T + 7T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
13 \( 1 - 0.272T + 13T^{2} \)
17 \( 1 - 3.49T + 17T^{2} \)
19 \( 1 - 3.02T + 19T^{2} \)
23 \( 1 - 1.09T + 23T^{2} \)
29 \( 1 + 7.94T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 7.55T + 41T^{2} \)
43 \( 1 + 9.00T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 9.15T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 2.02T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 + 9.97T + 97T^{2} \)
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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

−7.84831392049771438819760241469, −7.41231659080909398656662073924, −6.50918236484575791101407180839, −5.81626570277416935142769240493, −5.24856983620836348685527234000, −4.76066684077986375080011533680, −3.39235246659867595558425437609, −1.88862987138345478383213560440, −1.27890964253034573014933820530, 0, 1.27890964253034573014933820530, 1.88862987138345478383213560440, 3.39235246659867595558425437609, 4.76066684077986375080011533680, 5.24856983620836348685527234000, 5.81626570277416935142769240493, 6.50918236484575791101407180839, 7.41231659080909398656662073924, 7.84831392049771438819760241469

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