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 + 1.65·3-s + 4-s + 0.507·5-s − 1.65·6-s + 5.04·7-s − 8-s − 0.255·9-s − 0.507·10-s − 4.85·11-s + 1.65·12-s − 5.83·13-s − 5.04·14-s + 0.841·15-s + 16-s − 6.23·17-s + 0.255·18-s + 5.21·19-s + 0.507·20-s + 8.35·21-s + 4.85·22-s + 3.18·23-s − 1.65·24-s − 4.74·25-s + 5.83·26-s − 5.39·27-s + 5.04·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.956·3-s + 0.5·4-s + 0.227·5-s − 0.676·6-s + 1.90·7-s − 0.353·8-s − 0.0853·9-s − 0.160·10-s − 1.46·11-s + 0.478·12-s − 1.61·13-s − 1.34·14-s + 0.217·15-s + 0.250·16-s − 1.51·17-s + 0.0603·18-s + 1.19·19-s + 0.113·20-s + 1.82·21-s + 1.03·22-s + 0.663·23-s − 0.338·24-s − 0.948·25-s + 1.14·26-s − 1.03·27-s + 0.952·28-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 \( 1 - 1.65T + 3T^{2} \)
5 \( 1 - 0.507T + 5T^{2} \)
7 \( 1 - 5.04T + 7T^{2} \)
11 \( 1 + 4.85T + 11T^{2} \)
13 \( 1 + 5.83T + 13T^{2} \)
17 \( 1 + 6.23T + 17T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 - 3.18T + 23T^{2} \)
29 \( 1 + 3.02T + 29T^{2} \)
31 \( 1 + 4.77T + 31T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 - 2.07T + 41T^{2} \)
43 \( 1 + 0.260T + 43T^{2} \)
47 \( 1 - 2.71T + 47T^{2} \)
53 \( 1 - 7.62T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 + 5.34T + 67T^{2} \)
71 \( 1 + 5.34T + 71T^{2} \)
73 \( 1 + 5.71T + 73T^{2} \)
79 \( 1 - 4.02T + 79T^{2} \)
83 \( 1 + 6.95T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−8.049598264675190448210728393715, −7.47237650036464743500964457365, −7.28605633226472602318231799516, −5.61834819066423962069041728735, −5.16621596145135330475774567672, −4.35793615302213054079219606053, −2.98060028401611516936467933253, −2.27716299536455418539997113964, −1.76317761498612334516727913233, 0, 1.76317761498612334516727913233, 2.27716299536455418539997113964, 2.98060028401611516936467933253, 4.35793615302213054079219606053, 5.16621596145135330475774567672, 5.61834819066423962069041728735, 7.28605633226472602318231799516, 7.47237650036464743500964457365, 8.049598264675190448210728393715

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