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 + 1.76·3-s + 4-s − 4.09·5-s − 1.76·6-s + 4.14·7-s − 8-s + 0.112·9-s + 4.09·10-s − 5.57·11-s + 1.76·12-s + 2.12·13-s − 4.14·14-s − 7.21·15-s + 16-s + 3.96·17-s − 0.112·18-s − 8.26·19-s − 4.09·20-s + 7.32·21-s + 5.57·22-s + 6.53·23-s − 1.76·24-s + 11.7·25-s − 2.12·26-s − 5.09·27-s + 4.14·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.01·3-s + 0.5·4-s − 1.82·5-s − 0.720·6-s + 1.56·7-s − 0.353·8-s + 0.0375·9-s + 1.29·10-s − 1.68·11-s + 0.509·12-s + 0.590·13-s − 1.10·14-s − 1.86·15-s + 0.250·16-s + 0.961·17-s − 0.0265·18-s − 1.89·19-s − 0.914·20-s + 1.59·21-s + 1.18·22-s + 1.36·23-s − 0.360·24-s + 2.34·25-s − 0.417·26-s − 0.980·27-s + 0.784·28-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 \( 1 - 1.76T + 3T^{2} \)
5 \( 1 + 4.09T + 5T^{2} \)
7 \( 1 - 4.14T + 7T^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 - 2.12T + 13T^{2} \)
17 \( 1 - 3.96T + 17T^{2} \)
19 \( 1 + 8.26T + 19T^{2} \)
23 \( 1 - 6.53T + 23T^{2} \)
29 \( 1 + 0.348T + 29T^{2} \)
31 \( 1 - 5.91T + 31T^{2} \)
37 \( 1 - 2.19T + 37T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 + 4.05T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 9.45T + 67T^{2} \)
71 \( 1 + 4.62T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 8.81T + 83T^{2} \)
89 \( 1 + 1.84T + 89T^{2} \)
97 \( 1 + 8.67T + 97T^{2} \)
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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

−8.163646217118578614980857443831, −7.69517363339638909387083756394, −7.24992519202062504509305658594, −5.88472864934695398679788263654, −4.79968413792106251028854336945, −4.25881856706768784724345865667, −3.14859973698603925431763731460, −2.61285535633622267744538696272, −1.35832588137743795112430323026, 0, 1.35832588137743795112430323026, 2.61285535633622267744538696272, 3.14859973698603925431763731460, 4.25881856706768784724345865667, 4.79968413792106251028854336945, 5.88472864934695398679788263654, 7.24992519202062504509305658594, 7.69517363339638909387083756394, 8.163646217118578614980857443831

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