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 − 1.82·3-s + 4-s − 3.80·5-s + 1.82·6-s − 3.54·7-s − 8-s + 0.315·9-s + 3.80·10-s − 4.23·11-s − 1.82·12-s + 1.93·13-s + 3.54·14-s + 6.92·15-s + 16-s − 7.25·17-s − 0.315·18-s + 1.65·19-s − 3.80·20-s + 6.45·21-s + 4.23·22-s + 2.77·23-s + 1.82·24-s + 9.44·25-s − 1.93·26-s + 4.88·27-s − 3.54·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.05·3-s + 0.5·4-s − 1.69·5-s + 0.743·6-s − 1.33·7-s − 0.353·8-s + 0.105·9-s + 1.20·10-s − 1.27·11-s − 0.525·12-s + 0.535·13-s + 0.947·14-s + 1.78·15-s + 0.250·16-s − 1.75·17-s − 0.0744·18-s + 0.379·19-s − 0.849·20-s + 1.40·21-s + 0.903·22-s + 0.579·23-s + 0.371·24-s + 1.88·25-s − 0.378·26-s + 0.940·27-s − 0.669·28-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 \( 1 + 1.82T + 3T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 - 1.93T + 13T^{2} \)
17 \( 1 + 7.25T + 17T^{2} \)
19 \( 1 - 1.65T + 19T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 - 6.62T + 29T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
37 \( 1 + 1.32T + 37T^{2} \)
41 \( 1 - 7.72T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 - 2.77T + 47T^{2} \)
53 \( 1 + 8.53T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 2.85T + 71T^{2} \)
73 \( 1 - 5.54T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 9.05T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 3.33T + 97T^{2} \)
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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

−7.947218529264030819534182516361, −7.41849190131275264824858604583, −6.63777364396543384269564880493, −6.12919825383342650828542549011, −5.11623307771006073486135695495, −4.29093681380465449379583510807, −3.31977023282282205460793435115, −2.60791660910353681225428941729, −0.66056229024249714485810315341, 0, 0.66056229024249714485810315341, 2.60791660910353681225428941729, 3.31977023282282205460793435115, 4.29093681380465449379583510807, 5.11623307771006073486135695495, 6.12919825383342650828542549011, 6.63777364396543384269564880493, 7.41849190131275264824858604583, 7.947218529264030819534182516361

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