Properties

Degree 2
Conductor 4003
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.74·2-s − 0.797·3-s + 5.52·4-s − 1.01·5-s + 2.18·6-s − 0.805·7-s − 9.67·8-s − 2.36·9-s + 2.79·10-s − 3.88·11-s − 4.40·12-s − 6.48·13-s + 2.21·14-s + 0.812·15-s + 15.4·16-s + 4.70·17-s + 6.48·18-s + 0.952·19-s − 5.62·20-s + 0.642·21-s + 10.6·22-s + 5.52·23-s + 7.71·24-s − 3.96·25-s + 17.7·26-s + 4.27·27-s − 4.45·28-s + ⋯
L(s)  = 1  − 1.93·2-s − 0.460·3-s + 2.76·4-s − 0.455·5-s + 0.893·6-s − 0.304·7-s − 3.42·8-s − 0.787·9-s + 0.883·10-s − 1.17·11-s − 1.27·12-s − 1.79·13-s + 0.590·14-s + 0.209·15-s + 3.87·16-s + 1.14·17-s + 1.52·18-s + 0.218·19-s − 1.25·20-s + 0.140·21-s + 2.27·22-s + 1.15·23-s + 1.57·24-s − 0.792·25-s + 3.48·26-s + 0.823·27-s − 0.841·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4003,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02994355014$
$L(\frac12)$  $\approx$  $0.02994355014$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 4003$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 4003$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad4003 \( 1+O(T) \)
good2 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 + 0.797T + 3T^{2} \)
5 \( 1 + 1.01T + 5T^{2} \)
7 \( 1 + 0.805T + 7T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + 6.48T + 13T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 0.952T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 - 1.89T + 29T^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 + 8.60T + 37T^{2} \)
41 \( 1 + 9.32T + 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 - 1.56T + 47T^{2} \)
53 \( 1 - 0.105T + 53T^{2} \)
59 \( 1 + 7.49T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 7.18T + 79T^{2} \)
83 \( 1 + 4.55T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−8.410635372029805278107902756975, −7.77298117523728092262886052988, −7.29614584468330504909545031752, −6.66451614223486261920870223362, −5.52790751151155787915217638458, −5.15823037404109069431261974933, −3.23689496091631121012002781539, −2.77562727940343840634765061888, −1.65021773488463162241459106162, −0.12961446540553736700143344402, 0.12961446540553736700143344402, 1.65021773488463162241459106162, 2.77562727940343840634765061888, 3.23689496091631121012002781539, 5.15823037404109069431261974933, 5.52790751151155787915217638458, 6.66451614223486261920870223362, 7.29614584468330504909545031752, 7.77298117523728092262886052988, 8.410635372029805278107902756975

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