Properties

Degree $4$
Conductor $1444$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 3·3-s + 48·4-s − 45·5-s − 24·6-s + 114·7-s − 256·8-s − 119·9-s + 360·10-s + 661·11-s + 144·12-s + 1.61e3·13-s − 912·14-s − 135·15-s + 1.28e3·16-s + 64·17-s + 952·18-s + 722·19-s − 2.16e3·20-s + 342·21-s − 5.28e3·22-s − 3.18e3·23-s − 768·24-s − 1.48e3·25-s − 1.29e4·26-s − 12·27-s + 5.47e3·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.192·3-s + 3/2·4-s − 0.804·5-s − 0.272·6-s + 0.879·7-s − 1.41·8-s − 0.489·9-s + 1.13·10-s + 1.64·11-s + 0.288·12-s + 2.64·13-s − 1.24·14-s − 0.154·15-s + 5/4·16-s + 0.0537·17-s + 0.692·18-s + 0.458·19-s − 1.20·20-s + 0.169·21-s − 2.32·22-s − 1.25·23-s − 0.272·24-s − 0.476·25-s − 3.74·26-s − 0.00316·27-s + 1.31·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1444,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.26189\)
\(L(\frac12)\) \(\approx\) \(1.26189\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{2} T )^{2} \)
19$C_1$ \( ( 1 - p^{2} T )^{2} \)
good3$D_{4}$ \( 1 - p T + 128 T^{2} - p^{6} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 9 p T + 3514 T^{2} + 9 p^{6} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 - 114 T + 31099 T^{2} - 114 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 661 T + 430972 T^{2} - 661 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 1613 T + 1384022 T^{2} - 1613 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 64 T + 2804713 T^{2} - 64 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 3185 T + 14543782 T^{2} + 3185 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 2481 T + 6815332 T^{2} + 2481 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 1180 T + 21633278 T^{2} + 1180 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 10488 T + 155529814 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 16630 T + 170295586 T^{2} - 16630 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 11303 T + 297510638 T^{2} - 11303 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 12155 T + 47754214 T^{2} + 12155 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 20585 T + 812203882 T^{2} - 20585 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 78581 T + 2971672216 T^{2} + 78581 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 7805 T - 748756690 T^{2} - 7805 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 62488 T + 4005427642 T^{2} + 62488 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 16218 T + 1004140843 T^{2} - 16218 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 67122 T + 7273984870 T^{2} - 67122 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 10714 T + 5751433246 T^{2} + 10714 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 128188 T + 8689285330 T^{2} - 128188 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 178558 T + 22668743394 T^{2} - 178558 p^{5} T^{3} + p^{10} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.75426159982457036939000294038, −15.21649195166100810141507367349, −14.37729787115649742853691662155, −14.14719465065775536971790989698, −13.21687381214999115940021011408, −12.22143141025689843245357753400, −11.44928526567325200537447652435, −11.38162843749217152284541242011, −10.88283827358838766111517938128, −9.831642035698584421586795238501, −9.010431991225445523130268080765, −8.737522785476379508084403629336, −7.910684662506626978964080911307, −7.63645568361136861563448106403, −6.24183394782322365801246486384, −6.05181507622722288483448203211, −4.15004349959309532662649681447, −3.47819687037279433062730367506, −1.72764903657296812524078019374, −0.886440656638640508972368436622, 0.886440656638640508972368436622, 1.72764903657296812524078019374, 3.47819687037279433062730367506, 4.15004349959309532662649681447, 6.05181507622722288483448203211, 6.24183394782322365801246486384, 7.63645568361136861563448106403, 7.910684662506626978964080911307, 8.737522785476379508084403629336, 9.010431991225445523130268080765, 9.831642035698584421586795238501, 10.88283827358838766111517938128, 11.38162843749217152284541242011, 11.44928526567325200537447652435, 12.22143141025689843245357753400, 13.21687381214999115940021011408, 14.14719465065775536971790989698, 14.37729787115649742853691662155, 15.21649195166100810141507367349, 15.75426159982457036939000294038

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