Properties

Label 4-22e2-1.1-c9e2-0-1
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $128.386$
Root an. cond. $3.36612$
Motivic weight $9$
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  − 32·2-s + 34·3-s + 768·4-s − 1.47e3·5-s − 1.08e3·6-s + 8.19e3·7-s − 1.63e4·8-s − 8.86e3·9-s + 4.72e4·10-s − 2.92e4·11-s + 2.61e4·12-s − 1.96e5·13-s − 2.62e5·14-s − 5.02e4·15-s + 3.27e5·16-s − 5.48e5·17-s + 2.83e5·18-s − 9.09e4·19-s − 1.13e6·20-s + 2.78e5·21-s + 9.37e5·22-s − 3.97e6·23-s − 5.57e5·24-s + 6.95e5·25-s + 6.28e6·26-s + 2.69e4·27-s + 6.29e6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.242·3-s + 3/2·4-s − 1.05·5-s − 0.342·6-s + 1.29·7-s − 1.41·8-s − 0.450·9-s + 1.49·10-s − 0.603·11-s + 0.363·12-s − 1.90·13-s − 1.82·14-s − 0.256·15-s + 5/4·16-s − 1.59·17-s + 0.637·18-s − 0.160·19-s − 1.58·20-s + 0.312·21-s + 0.852·22-s − 2.95·23-s − 0.342·24-s + 0.356·25-s + 2.69·26-s + 0.00976·27-s + 1.93·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(128.386\)
Root analytic conductor: \(3.36612\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 484,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{2} \)
11$C_1$ \( ( 1 + p^{4} T )^{2} \)
good3$D_{4}$ \( 1 - 34 T + 3341 p T^{2} - 34 p^{9} T^{3} + p^{18} T^{4} \)
5$D_{4}$ \( 1 + 1478 T + 1489171 T^{2} + 1478 p^{9} T^{3} + p^{18} T^{4} \)
7$D_{4}$ \( 1 - 8196 T + 13416478 p T^{2} - 8196 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 + 196296 T + 29063367850 T^{2} + 196296 p^{9} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 + 548788 T + 307702964998 T^{2} + 548788 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 + 90928 T + 592554394982 T^{2} + 90928 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 + 3971046 T + 7439094199447 T^{2} + 3971046 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 656128 T + 29120660476282 T^{2} + 656128 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 + 872214 T + 31353947173879 T^{2} + 872214 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 + 190722 T + 102568285144843 T^{2} + 190722 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 3350080 T - 140375144937710 T^{2} - 3350080 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 - 3876012 T + 465211326598522 T^{2} - 3876012 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 - 38884544 T + 2616247440040030 T^{2} - 38884544 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 + 134346316 T + 10740733254941902 T^{2} + 134346316 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 + 125273754 T + 13934210655422239 T^{2} + 125273754 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 114821880 T + 26575222920085450 T^{2} - 114821880 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 + 91519714 T - 5419714607912857 T^{2} + 91519714 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 - 411397438 T + 122021318156477455 T^{2} - 411397438 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 - 506142392 T + 178059740401953890 T^{2} - 506142392 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 - 810072516 T + 399258205296441634 T^{2} - 810072516 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 + 718904884 T + 445583716474989322 T^{2} + 718904884 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 + 810786322 T + 689466379074066307 T^{2} + 810786322 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 + 230208654 T + 1520560882324892275 T^{2} + 230208654 p^{9} T^{3} + p^{18} T^{4} \)
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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

−15.56569156908112728706651063278, −15.26728600385056577542467715292, −14.38205586195253711362988505677, −13.99631597193852905423893990863, −12.43935545131287576210859686348, −12.17409670454820283536462162163, −11.15685180452479996982594280804, −11.09388809417739378899833019315, −9.985890223518318507375689218998, −9.363656615848476109924289887953, −8.237131069032866427488817951059, −8.044394305231631791141588211627, −7.50261571657642843043288788298, −6.44316847451934672493812962566, −5.08052989551121481348464565365, −4.12074172629341009915260466555, −2.49930774447671632607690690401, −1.90124099209667037110766883982, 0, 0, 1.90124099209667037110766883982, 2.49930774447671632607690690401, 4.12074172629341009915260466555, 5.08052989551121481348464565365, 6.44316847451934672493812962566, 7.50261571657642843043288788298, 8.044394305231631791141588211627, 8.237131069032866427488817951059, 9.363656615848476109924289887953, 9.985890223518318507375689218998, 11.09388809417739378899833019315, 11.15685180452479996982594280804, 12.17409670454820283536462162163, 12.43935545131287576210859686348, 13.99631597193852905423893990863, 14.38205586195253711362988505677, 15.26728600385056577542467715292, 15.56569156908112728706651063278

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