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 + 1.40·3-s + 4-s − 2.54·5-s − 1.40·6-s − 0.597·7-s − 8-s − 1.03·9-s + 2.54·10-s − 1.71·11-s + 1.40·12-s + 3.98·13-s + 0.597·14-s − 3.57·15-s + 16-s − 6.84·17-s + 1.03·18-s + 5.26·19-s − 2.54·20-s − 0.836·21-s + 1.71·22-s + 4.99·23-s − 1.40·24-s + 1.50·25-s − 3.98·26-s − 5.65·27-s − 0.597·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.808·3-s + 0.5·4-s − 1.14·5-s − 0.571·6-s − 0.225·7-s − 0.353·8-s − 0.346·9-s + 0.806·10-s − 0.517·11-s + 0.404·12-s + 1.10·13-s + 0.159·14-s − 0.921·15-s + 0.250·16-s − 1.66·17-s + 0.244·18-s + 1.20·19-s − 0.570·20-s − 0.182·21-s + 0.365·22-s + 1.04·23-s − 0.285·24-s + 0.300·25-s − 0.782·26-s − 1.08·27-s − 0.112·28-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 \( 1 - 1.40T + 3T^{2} \)
5 \( 1 + 2.54T + 5T^{2} \)
7 \( 1 + 0.597T + 7T^{2} \)
11 \( 1 + 1.71T + 11T^{2} \)
13 \( 1 - 3.98T + 13T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
19 \( 1 - 5.26T + 19T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 - 9.98T + 31T^{2} \)
37 \( 1 - 5.28T + 37T^{2} \)
41 \( 1 + 1.39T + 41T^{2} \)
43 \( 1 + 7.08T + 43T^{2} \)
47 \( 1 - 0.390T + 47T^{2} \)
53 \( 1 + 3.51T + 53T^{2} \)
59 \( 1 + 8.12T + 59T^{2} \)
61 \( 1 - 1.54T + 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 6.99T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 3.92T + 83T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.290846646910620021242064223756, −7.64206362606191134546459121620, −6.79522930268213118775709640816, −6.16035095083463268306329606636, −4.93804394037277463490935513582, −4.09741917380040761307575219360, −3.07367948322020849310428172076, −2.75633528973116516444856619845, −1.27522767943611782804032829108, 0, 1.27522767943611782804032829108, 2.75633528973116516444856619845, 3.07367948322020849310428172076, 4.09741917380040761307575219360, 4.93804394037277463490935513582, 6.16035095083463268306329606636, 6.79522930268213118775709640816, 7.64206362606191134546459121620, 8.290846646910620021242064223756

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