Properties

Label 4-21e2-1.1-c7e2-0-2
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s − 54·3-s + 71·4-s − 360·5-s + 486·6-s + 686·7-s − 1.70e3·8-s + 2.18e3·9-s + 3.24e3·10-s − 4.93e3·11-s − 3.83e3·12-s + 7.70e3·13-s − 6.17e3·14-s + 1.94e4·15-s + 8.58e3·16-s − 2.85e4·17-s − 1.96e4·18-s − 6.37e4·19-s − 2.55e4·20-s − 3.70e4·21-s + 4.43e4·22-s + 8.22e4·23-s + 9.18e4·24-s + 4.74e4·25-s − 6.93e4·26-s − 7.87e4·27-s + 4.87e4·28-s + ⋯
L(s)  = 1  − 0.795·2-s − 1.15·3-s + 0.554·4-s − 1.28·5-s + 0.918·6-s + 0.755·7-s − 1.17·8-s + 9-s + 1.02·10-s − 1.11·11-s − 0.640·12-s + 0.973·13-s − 0.601·14-s + 1.48·15-s + 0.523·16-s − 1.41·17-s − 0.795·18-s − 2.13·19-s − 0.714·20-s − 0.872·21-s + 0.888·22-s + 1.40·23-s + 1.35·24-s + 0.607·25-s − 0.774·26-s − 0.769·27-s + 0.419·28-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: \(2\)
Selberg data: \((4,\ 441,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$D_{4}$ \( 1 + 9 T + 5 p T^{2} + 9 p^{7} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 + 72 p T + 3286 p^{2} T^{2} + 72 p^{8} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 + 4932 T + 19797958 T^{2} + 4932 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 - 7708 T + 118266510 T^{2} - 7708 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 28584 T + 942631150 T^{2} + 28584 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 63728 T + 2569797414 T^{2} + 63728 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 82260 T + 8202169294 T^{2} - 82260 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 435996 T + 80862098782 T^{2} + 435996 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 + 29240 T + 27798421182 T^{2} + 29240 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 709556 T + 313296440190 T^{2} + 709556 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 25056 T - 26592839954 T^{2} + 25056 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 496216 T + 567160908438 T^{2} - 496216 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 1575000 T + 1564490669086 T^{2} + 1575000 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 2057436 T + 3149566808638 T^{2} - 2057436 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 1101024 T + 4521593481622 T^{2} + 1101024 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 - 28996 T + 5887677435486 T^{2} - 28996 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 4480784 T + 16777543143750 T^{2} + 4480784 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 54540 T + 17803030672942 T^{2} - 54540 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 666604 T - 310686963642 T^{2} - 666604 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 2322952 T + 38986658128734 T^{2} - 2322952 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 7384392 T + 61089776413510 T^{2} + 7384392 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 1784448 T + 26626978018894 T^{2} - 1784448 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 16266412 T + 223770781220502 T^{2} - 16266412 p^{7} T^{3} + p^{14} T^{4} \)
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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

−16.35376414830336104329946876990, −15.49353920862973173186092969531, −15.33906116366350644198804511681, −14.85134476712614553392459284003, −13.11353841118754649164283965869, −12.98702501814532961441949753557, −11.93048316411985654992076868186, −11.19125360556158486390801704942, −11.06864907834849141420315822462, −10.48357678196108462687760349283, −8.923696557726720156659706900536, −8.662300618383784066586513339856, −7.59432258364094413923615516970, −6.88728593278890824351320425883, −5.91638510993332172401398668443, −4.84819502217452942086127117110, −3.76338125254263880029545162404, −1.94579027207014780314016043593, 0, 0, 1.94579027207014780314016043593, 3.76338125254263880029545162404, 4.84819502217452942086127117110, 5.91638510993332172401398668443, 6.88728593278890824351320425883, 7.59432258364094413923615516970, 8.662300618383784066586513339856, 8.923696557726720156659706900536, 10.48357678196108462687760349283, 11.06864907834849141420315822462, 11.19125360556158486390801704942, 11.93048316411985654992076868186, 12.98702501814532961441949753557, 13.11353841118754649164283965869, 14.85134476712614553392459284003, 15.33906116366350644198804511681, 15.49353920862973173186092969531, 16.35376414830336104329946876990

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