Properties

Label 4-21e2-1.1-c7e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $43.0347$
Root an. cond. $2.56126$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·2-s − 54·3-s + 120·4-s − 24·5-s − 648·6-s − 686·7-s + 2.68e3·8-s + 2.18e3·9-s − 288·10-s + 2.12e3·11-s − 6.48e3·12-s − 1.08e3·13-s − 8.23e3·14-s + 1.29e3·15-s + 3.14e4·16-s − 2.92e4·17-s + 2.62e4·18-s − 2.58e4·19-s − 2.88e3·20-s + 3.70e4·21-s + 2.54e4·22-s + 6.83e4·23-s − 1.45e5·24-s − 1.38e5·25-s − 1.30e4·26-s − 7.87e4·27-s − 8.23e4·28-s + ⋯
L(s)  = 1  + 1.06·2-s − 1.15·3-s + 0.937·4-s − 0.0858·5-s − 1.22·6-s − 0.755·7-s + 1.85·8-s + 9-s − 0.0910·10-s + 0.481·11-s − 1.08·12-s − 0.136·13-s − 0.801·14-s + 0.0991·15-s + 1.91·16-s − 1.44·17-s + 1.06·18-s − 0.863·19-s − 0.0804·20-s + 0.872·21-s + 0.510·22-s + 1.17·23-s − 2.14·24-s − 1.77·25-s − 0.145·26-s − 0.769·27-s − 0.708·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(43.0347\)
Root analytic conductor: \(2.56126\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.743830273\)
\(L(\frac12)\) \(\approx\) \(2.743830273\)
\(L(\frac{9}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{3} T )^{2} \)
7$C_1$ \( ( 1 + p^{3} T )^{2} \)
good2$D_{4}$ \( 1 - 3 p^{2} T + 3 p^{3} T^{2} - 3 p^{9} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 + 24 T + 139242 T^{2} + 24 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 2124 T + 19090986 T^{2} - 2124 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 1084 T + 103561806 T^{2} + 1084 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 29256 T + 533114098 T^{2} + 29256 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 25816 T + 1627717254 T^{2} + 25816 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 68316 T + 7976265490 T^{2} - 68316 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 211308 T + 35807598606 T^{2} - 211308 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 435840 T + 98660711870 T^{2} - 435840 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 28428 T + 188976571454 T^{2} + 28428 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 749760 T + 517210006962 T^{2} - 749760 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 397096 T + 472051884246 T^{2} - 397096 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 840168 T + 744642372910 T^{2} - 840168 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 246684 T + 2085795052990 T^{2} + 246684 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 2199504 T + 4631611593574 T^{2} - 2199504 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 1951108 T + 6790017439086 T^{2} + 1951108 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 1532048 T + 530136191190 T^{2} - 1532048 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 2024004 T + 14260375775986 T^{2} - 2024004 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 1709028 T + 11306816102198 T^{2} + 1709028 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 1048168 T + 11630316806382 T^{2} - 1048168 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 4894296 T + 47206286808070 T^{2} + 4894296 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 + 60864 T + 81562257471570 T^{2} + 60864 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 26046852 T + 325711749428294 T^{2} + 26046852 p^{7} T^{3} + p^{14} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.88224499247058652691081534702, −16.02811316544607560981621295880, −15.70250906837755466712199071171, −15.18851969542109728930663728516, −13.95625076943507760994215666074, −13.65206000825996643500787096170, −12.89709641092255898090487161303, −12.35402637275694830083805626076, −11.63743149679453847438288939971, −10.97429203686337127180376045665, −10.41394527918625825593600050991, −9.613869564836525851004311774933, −8.264832141994961435531319042294, −7.09346540608775711954501655338, −6.54530664648629794277838584581, −5.82581005654895952626718374074, −4.51729838368586493760445633859, −4.24535835091368686150635760271, −2.46334465876767302777906787572, −0.885108843589874238964748372576, 0.885108843589874238964748372576, 2.46334465876767302777906787572, 4.24535835091368686150635760271, 4.51729838368586493760445633859, 5.82581005654895952626718374074, 6.54530664648629794277838584581, 7.09346540608775711954501655338, 8.264832141994961435531319042294, 9.613869564836525851004311774933, 10.41394527918625825593600050991, 10.97429203686337127180376045665, 11.63743149679453847438288939971, 12.35402637275694830083805626076, 12.89709641092255898090487161303, 13.65206000825996643500787096170, 13.95625076943507760994215666074, 15.18851969542109728930663728516, 15.70250906837755466712199071171, 16.02811316544607560981621295880, 16.88224499247058652691081534702

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