Properties

Label 4-22e2-1.1-c13e2-0-2
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $556.526$
Root an. cond. $4.85703$
Motivic weight $13$
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  + 128·2-s − 926·3-s + 1.22e4·4-s + 2.91e3·5-s − 1.18e5·6-s − 1.70e5·7-s + 1.04e6·8-s − 2.36e6·9-s + 3.72e5·10-s + 3.54e6·11-s − 1.13e7·12-s − 6.94e6·13-s − 2.18e7·14-s − 2.69e6·15-s + 8.38e7·16-s − 2.80e8·17-s − 3.02e8·18-s − 3.49e8·19-s + 3.58e7·20-s + 1.57e8·21-s + 4.53e8·22-s − 1.51e8·23-s − 9.70e8·24-s − 2.01e9·25-s − 8.89e8·26-s + 3.69e9·27-s − 2.09e9·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.733·3-s + 3/2·4-s + 0.0834·5-s − 1.03·6-s − 0.547·7-s + 1.41·8-s − 1.48·9-s + 0.117·10-s + 0.603·11-s − 1.10·12-s − 0.399·13-s − 0.774·14-s − 0.0611·15-s + 5/4·16-s − 2.81·17-s − 2.09·18-s − 1.70·19-s + 0.125·20-s + 0.401·21-s + 0.852·22-s − 0.213·23-s − 1.03·24-s − 1.64·25-s − 0.564·26-s + 1.83·27-s − 0.821·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(556.526\)
Root analytic conductor: \(4.85703\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 484,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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$C_1$ \( ( 1 - p^{6} T )^{2} \)
11$C_1$ \( ( 1 - p^{6} T )^{2} \)
good3$D_{4}$ \( 1 + 926 T + 119257 p^{3} T^{2} + 926 p^{13} T^{3} + p^{26} T^{4} \)
5$D_{4}$ \( 1 - 2914 T + 404344399 p T^{2} - 2914 p^{13} T^{3} + p^{26} T^{4} \)
7$D_{4}$ \( 1 + 170560 T - 43177261686 T^{2} + 170560 p^{13} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 + 534532 p T + 481527257590170 T^{2} + 534532 p^{14} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 + 280144288 T + 39297996620745566 T^{2} + 280144288 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 349231788 T + 76902832929625670 T^{2} + 349231788 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 + 151278294 T + 176428020500937379 T^{2} + 151278294 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 + 771621928 T + 12702434689985788058 T^{2} + 771621928 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 + 8626482070 T + 63994786766637916563 T^{2} + 8626482070 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 + 26333898490 T + \)\(61\!\cdots\!55\)\( T^{2} + 26333898490 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 + 3560991836 T - 43407112458636956110 T^{2} + 3560991836 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 + 16329995932 T - \)\(11\!\cdots\!34\)\( T^{2} + 16329995932 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 - 9513351704 T + \)\(10\!\cdots\!42\)\( T^{2} - 9513351704 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 - 291363898652 T + \)\(50\!\cdots\!46\)\( T^{2} - 291363898652 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 - 712019011182 T + \)\(33\!\cdots\!15\)\( T^{2} - 712019011182 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 - 138918582944 T + \)\(43\!\cdots\!02\)\( T^{2} - 138918582944 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 - 912599195574 T + \)\(12\!\cdots\!67\)\( T^{2} - 912599195574 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 + 1560848343722 T + \)\(27\!\cdots\!27\)\( T^{2} + 1560848343722 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 - 407382417996 T + \)\(28\!\cdots\!14\)\( T^{2} - 407382417996 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 + 48102676468 T + \)\(78\!\cdots\!58\)\( T^{2} + 48102676468 p^{13} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 + 6313820551012 T + \)\(25\!\cdots\!62\)\( T^{2} + 6313820551012 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 - 977692325462 T + \)\(14\!\cdots\!75\)\( T^{2} - 977692325462 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 + 1024977303502 T + \)\(10\!\cdots\!31\)\( T^{2} + 1024977303502 p^{13} T^{3} + p^{26} T^{4} \)
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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

−14.47260950304990092159756643032, −13.93029033954800279197859653898, −13.19955301172427507670759476744, −12.80465430124702030378719558401, −11.87056847538732148920548299586, −11.49628265895317265679347035662, −10.96288619234369241675564907812, −10.27185880055690870880260609990, −8.932834494770729573072928044959, −8.545791146511336848880689499103, −6.96619912073138807011229473611, −6.62189712472088249288249369941, −5.82732807123817702441251559464, −5.33849960843245318156815089537, −4.23589921813306534780070034085, −3.72994131632268153740503093360, −2.38462418729502696730585687076, −2.08076144442039860987026229816, 0, 0, 2.08076144442039860987026229816, 2.38462418729502696730585687076, 3.72994131632268153740503093360, 4.23589921813306534780070034085, 5.33849960843245318156815089537, 5.82732807123817702441251559464, 6.62189712472088249288249369941, 6.96619912073138807011229473611, 8.545791146511336848880689499103, 8.932834494770729573072928044959, 10.27185880055690870880260609990, 10.96288619234369241675564907812, 11.49628265895317265679347035662, 11.87056847538732148920548299586, 12.80465430124702030378719558401, 13.19955301172427507670759476744, 13.93029033954800279197859653898, 14.47260950304990092159756643032

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