Properties

Degree 4
Conductor $ 4003^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·7-s − 4·9-s − 4·11-s + 8·13-s − 2·17-s + 2·19-s − 12·23-s − 8·25-s + 4·28-s − 8·29-s + 4·31-s + 8·36-s − 14·37-s + 16·41-s + 8·44-s + 8·47-s − 11·49-s − 16·52-s − 4·53-s − 12·59-s + 20·61-s + 8·63-s + 8·64-s + 14·67-s + 4·68-s + 12·71-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s − 4/3·9-s − 1.20·11-s + 2.21·13-s − 0.485·17-s + 0.458·19-s − 2.50·23-s − 8/5·25-s + 0.755·28-s − 1.48·29-s + 0.718·31-s + 4/3·36-s − 2.30·37-s + 2.49·41-s + 1.20·44-s + 1.16·47-s − 1.57·49-s − 2.21·52-s − 0.549·53-s − 1.56·59-s + 2.56·61-s + 1.00·63-s + 64-s + 1.71·67-s + 0.485·68-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16024009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16024009 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16024009\)    =    \(4003^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4003} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 16024009,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 4003$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 4003$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad4003 \( 1+O(T) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 151 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 251 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 212 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.197238268154793147296299303082, −8.151364239265583733926626653248, −7.63008765758379215224417701137, −7.38015678836174200313665146198, −6.55663777766642901951761733324, −6.26011756104634778170814493927, −6.07297894844572667204275783056, −5.70019151937449400557254896563, −5.26053723403507263980916319769, −5.13954118223872036146552147011, −4.23832529337291424040604495150, −4.01290259266816417020876944909, −3.56605559268132397838855381812, −3.50585943082748260518053743449, −2.71798638211430985859676953885, −2.18466034537402874734904992469, −1.90092769902051903625201955119, −0.898079468955872573049789736613, 0, 0, 0.898079468955872573049789736613, 1.90092769902051903625201955119, 2.18466034537402874734904992469, 2.71798638211430985859676953885, 3.50585943082748260518053743449, 3.56605559268132397838855381812, 4.01290259266816417020876944909, 4.23832529337291424040604495150, 5.13954118223872036146552147011, 5.26053723403507263980916319769, 5.70019151937449400557254896563, 6.07297894844572667204275783056, 6.26011756104634778170814493927, 6.55663777766642901951761733324, 7.38015678836174200313665146198, 7.63008765758379215224417701137, 8.151364239265583733926626653248, 8.197238268154793147296299303082

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