Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 1.41·3-s − 1.41·5-s + 2.00·6-s − 7-s + 2.82·8-s − 0.999·9-s + 2.00·10-s + 0.828·11-s + 1.17·13-s + 1.41·14-s + 2.00·15-s − 4.00·16-s + 4.65·17-s + 1.41·18-s − 4.65·19-s + 1.41·21-s − 1.17·22-s − 8.82·23-s − 4·24-s − 2.99·25-s − 1.65·26-s + 5.65·27-s − 1.17·29-s − 2.82·30-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.816·3-s − 0.632·5-s + 0.816·6-s − 0.377·7-s + 0.999·8-s − 0.333·9-s + 0.632·10-s + 0.249·11-s + 0.324·13-s + 0.377·14-s + 0.516·15-s − 1.00·16-s + 1.12·17-s + 0.333·18-s − 1.06·19-s + 0.308·21-s − 0.249·22-s − 1.84·23-s − 0.816·24-s − 0.599·25-s − 0.324·26-s + 1.08·27-s − 0.217·29-s − 0.516·30-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  =  1
Selberg data  =  $(2,\ 4003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 4003$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 4003$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad4003 \( 1+O(T) \)
good2 \( 1 + 1.41T + 2T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 - 4.65T + 17T^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 + 8.82T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 4.17T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 - 7.17T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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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

−8.197238268154793147296299303082, −7.63008765758379215224417701137, −6.55663777766642901951761733324, −6.07297894844572667204275783056, −5.13954118223872036146552147011, −4.23832529337291424040604495150, −3.56605559268132397838855381812, −2.18466034537402874734904992469, −0.898079468955872573049789736613, 0, 0.898079468955872573049789736613, 2.18466034537402874734904992469, 3.56605559268132397838855381812, 4.23832529337291424040604495150, 5.13954118223872036146552147011, 6.07297894844572667204275783056, 6.55663777766642901951761733324, 7.63008765758379215224417701137, 8.197238268154793147296299303082

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