Properties

Label 4-22e2-1.1-c13e2-0-1
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 − 662·3-s + 1.22e4·4-s − 4.85e4·5-s + 8.47e4·6-s + 4.04e5·7-s − 1.04e6·8-s − 1.25e6·9-s + 6.21e6·10-s − 3.54e6·11-s − 8.13e6·12-s + 2.84e7·13-s − 5.17e7·14-s + 3.21e7·15-s + 8.38e7·16-s + 1.07e8·17-s + 1.61e8·18-s − 7.35e7·19-s − 5.96e8·20-s − 2.67e8·21-s + 4.53e8·22-s − 9.87e8·23-s + 6.94e8·24-s + 4.09e8·25-s − 3.64e9·26-s + 9.02e8·27-s + 4.97e9·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.524·3-s + 3/2·4-s − 1.39·5-s + 0.741·6-s + 1.29·7-s − 1.41·8-s − 0.789·9-s + 1.96·10-s − 0.603·11-s − 0.786·12-s + 1.63·13-s − 1.83·14-s + 0.728·15-s + 5/4·16-s + 1.07·17-s + 1.11·18-s − 0.358·19-s − 2.08·20-s − 0.681·21-s + 0.852·22-s − 1.39·23-s + 0.741·24-s + 0.335·25-s − 2.31·26-s + 0.448·27-s + 1.94·28-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$D_{4}$ \( 1 + 662 T + 565861 p T^{2} + 662 p^{13} T^{3} + p^{26} T^{4} \)
5$D_{4}$ \( 1 + 48566 T + 389809703 p T^{2} + 48566 p^{13} T^{3} + p^{26} T^{4} \)
7$D_{4}$ \( 1 - 404508 T + 16233930190 p T^{2} - 404508 p^{13} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 - 28445952 T + 720464142541066 T^{2} - 28445952 p^{13} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 - 107032052 T + 20369477483479894 T^{2} - 107032052 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 73524568 T + 79701138832491110 T^{2} + 73524568 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 + 987164862 T + 1209740889736512703 T^{2} + 987164862 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 + 9516399376 T + 42288055667555494426 T^{2} + 9516399376 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 + 6865940430 T + 59716769884901551807 T^{2} + 6865940430 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 + 4145233266 T + \)\(48\!\cdots\!59\)\( T^{2} + 4145233266 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 - 39451523656 T + \)\(18\!\cdots\!10\)\( T^{2} - 39451523656 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 - 5532549228 T + \)\(33\!\cdots\!86\)\( T^{2} - 5532549228 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 + 28411609744 T + \)\(11\!\cdots\!74\)\( T^{2} + 28411609744 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 + 49178542972 T + \)\(52\!\cdots\!42\)\( T^{2} + 49178542972 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 - 110326136718 T + \)\(20\!\cdots\!95\)\( T^{2} - 110326136718 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 + 161816895480 T + \)\(19\!\cdots\!62\)\( T^{2} + 161816895480 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 + 483661746106 T + \)\(11\!\cdots\!27\)\( T^{2} + 483661746106 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 - 318713704774 T + \)\(29\!\cdots\!95\)\( T^{2} - 318713704774 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 + 1073843539168 T + \)\(30\!\cdots\!46\)\( T^{2} + 1073843539168 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 + 742108796220 T + \)\(93\!\cdots\!22\)\( T^{2} + 742108796220 p^{13} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 + 6905249437156 T + \)\(29\!\cdots\!86\)\( T^{2} + 6905249437156 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 + 8233927751362 T + \)\(60\!\cdots\!63\)\( T^{2} + 8233927751362 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 + 4680888780654 T + \)\(18\!\cdots\!83\)\( T^{2} + 4680888780654 p^{13} T^{3} + p^{26} T^{4} \)
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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

−14.68450865446069579800254693425, −14.34143371740561674943008312177, −13.07532198773157781640708994647, −12.24105934152496135459761733453, −11.40686653329259410070933654687, −11.24607953926575255725377352193, −10.93230808130303376214251103328, −9.903069722480950904687456949573, −8.832228305756961803452375794937, −8.374812140758873910276229881054, −7.57798586955825398260201795273, −7.56321406342291859483157578567, −5.86240469474208148671174793232, −5.63741494300388003850907786446, −4.08304624777606676426899443657, −3.40193623511112733967665197024, −1.97759518356964779128479875291, −1.27910233354001605159055504260, 0, 0, 1.27910233354001605159055504260, 1.97759518356964779128479875291, 3.40193623511112733967665197024, 4.08304624777606676426899443657, 5.63741494300388003850907786446, 5.86240469474208148671174793232, 7.56321406342291859483157578567, 7.57798586955825398260201795273, 8.374812140758873910276229881054, 8.832228305756961803452375794937, 9.903069722480950904687456949573, 10.93230808130303376214251103328, 11.24607953926575255725377352193, 11.40686653329259410070933654687, 12.24105934152496135459761733453, 13.07532198773157781640708994647, 14.34143371740561674943008312177, 14.68450865446069579800254693425

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