Properties

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

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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