Properties

Label 4-22e2-1.1-c19e2-0-0
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $2534.08$
Root an. cond. $7.09504$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·2-s + 1.69e4·3-s + 7.86e5·4-s − 2.94e6·5-s + 1.73e7·6-s + 8.49e7·7-s + 5.36e8·8-s − 2.04e9·9-s − 3.01e9·10-s − 4.71e9·11-s + 1.33e10·12-s + 1.29e10·13-s + 8.69e10·14-s − 5.00e10·15-s + 3.43e11·16-s − 4.59e11·17-s − 2.09e12·18-s − 1.76e12·19-s − 2.31e12·20-s + 1.44e12·21-s − 4.82e12·22-s − 3.26e12·23-s + 9.11e12·24-s − 1.51e13·25-s + 1.32e13·26-s − 5.44e13·27-s + 6.67e13·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.497·3-s + 3/2·4-s − 0.675·5-s + 0.704·6-s + 0.795·7-s + 1.41·8-s − 1.75·9-s − 0.954·10-s − 0.603·11-s + 0.746·12-s + 0.338·13-s + 1.12·14-s − 0.336·15-s + 5/4·16-s − 0.940·17-s − 2.48·18-s − 1.25·19-s − 1.01·20-s + 0.396·21-s − 0.852·22-s − 0.378·23-s + 0.704·24-s − 0.793·25-s + 0.478·26-s − 1.37·27-s + 1.19·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2534.08\)
Root analytic conductor: \(7.09504\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 484,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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$C_1$ \( ( 1 - p^{9} T )^{2} \)
11$C_1$ \( ( 1 + p^{9} T )^{2} \)
good3$D_{4}$ \( 1 - 1886 p^{2} T + 9589549 p^{5} T^{2} - 1886 p^{21} T^{3} + p^{38} T^{4} \)
5$D_{4}$ \( 1 + 589702 p T + 952842082387 p^{2} T^{2} + 589702 p^{20} T^{3} + p^{38} T^{4} \)
7$D_{4}$ \( 1 - 84929956 T + 3178227775648110 p T^{2} - 84929956 p^{19} T^{3} + p^{38} T^{4} \)
13$D_{4}$ \( 1 - 12931357856 T + 1385574858552882338 p T^{2} - 12931357856 p^{19} T^{3} + p^{38} T^{4} \)
17$D_{4}$ \( 1 + 459787304876 T + \)\(29\!\cdots\!58\)\( p T^{2} + 459787304876 p^{19} T^{3} + p^{38} T^{4} \)
19$D_{4}$ \( 1 + 1766628927912 T + \)\(44\!\cdots\!10\)\( T^{2} + 1766628927912 p^{19} T^{3} + p^{38} T^{4} \)
23$D_{4}$ \( 1 + 3266600096146 T + \)\(30\!\cdots\!67\)\( T^{2} + 3266600096146 p^{19} T^{3} + p^{38} T^{4} \)
29$D_{4}$ \( 1 + 3348742635216 p T + \)\(14\!\cdots\!46\)\( T^{2} + 3348742635216 p^{20} T^{3} + p^{38} T^{4} \)
31$D_{4}$ \( 1 + 4975942037230 p T + \)\(46\!\cdots\!67\)\( T^{2} + 4975942037230 p^{20} T^{3} + p^{38} T^{4} \)
37$D_{4}$ \( 1 + 571392764844202 T + \)\(10\!\cdots\!31\)\( T^{2} + 571392764844202 p^{19} T^{3} + p^{38} T^{4} \)
41$D_{4}$ \( 1 + 184921680043224 T + \)\(83\!\cdots\!10\)\( T^{2} + 184921680043224 p^{19} T^{3} + p^{38} T^{4} \)
43$D_{4}$ \( 1 - 2116703700483204 T + \)\(52\!\cdots\!98\)\( p T^{2} - 2116703700483204 p^{19} T^{3} + p^{38} T^{4} \)
47$D_{4}$ \( 1 + 7456201298413488 T + \)\(85\!\cdots\!46\)\( T^{2} + 7456201298413488 p^{19} T^{3} + p^{38} T^{4} \)
53$D_{4}$ \( 1 + 14846639027241356 T + \)\(41\!\cdots\!18\)\( T^{2} + 14846639027241356 p^{19} T^{3} + p^{38} T^{4} \)
59$D_{4}$ \( 1 - 27278259292488522 T - \)\(37\!\cdots\!05\)\( T^{2} - 27278259292488522 p^{19} T^{3} + p^{38} T^{4} \)
61$D_{4}$ \( 1 + 221405767994019480 T + \)\(28\!\cdots\!82\)\( T^{2} + 221405767994019480 p^{19} T^{3} + p^{38} T^{4} \)
67$D_{4}$ \( 1 + 211824895464439422 T + \)\(10\!\cdots\!63\)\( T^{2} + 211824895464439422 p^{19} T^{3} + p^{38} T^{4} \)
71$D_{4}$ \( 1 + 752788827699476886 T + \)\(40\!\cdots\!95\)\( T^{2} + 752788827699476886 p^{19} T^{3} + p^{38} T^{4} \)
73$D_{4}$ \( 1 + 1171002362670305344 T + \)\(83\!\cdots\!94\)\( T^{2} + 1171002362670305344 p^{19} T^{3} + p^{38} T^{4} \)
79$D_{4}$ \( 1 - 1558374634535268220 T + \)\(24\!\cdots\!62\)\( T^{2} - 1558374634535268220 p^{19} T^{3} + p^{38} T^{4} \)
83$D_{4}$ \( 1 - 974129749376372 T + \)\(46\!\cdots\!34\)\( T^{2} - 974129749376372 p^{19} T^{3} + p^{38} T^{4} \)
89$D_{4}$ \( 1 - 3903208985507920622 T + \)\(13\!\cdots\!43\)\( T^{2} - 3903208985507920622 p^{19} T^{3} + p^{38} T^{4} \)
97$D_{4}$ \( 1 - 17667658647655093842 T + \)\(16\!\cdots\!07\)\( T^{2} - 17667658647655093842 p^{19} T^{3} + p^{38} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.13549784945325282811832403309, −13.11144064655754888494314970898, −11.88744328713633877464966356143, −11.59787809052696885108933576767, −10.98962583940140801249009851793, −10.57534600791652591081054214677, −9.114605605103133903707326114776, −8.559961584009904597206884504052, −7.85360194383042896060463463546, −7.41358897846472370213448914498, −6.07794087709443950223656155514, −5.93445064745839213776139501967, −4.89512376274024675056818790019, −4.40075203013966104219670295664, −3.47156669711153178032864462647, −3.11796591301443983081695124160, −1.99212186640714478474030902890, −1.91583052414582099047932367837, 0, 0, 1.91583052414582099047932367837, 1.99212186640714478474030902890, 3.11796591301443983081695124160, 3.47156669711153178032864462647, 4.40075203013966104219670295664, 4.89512376274024675056818790019, 5.93445064745839213776139501967, 6.07794087709443950223656155514, 7.41358897846472370213448914498, 7.85360194383042896060463463546, 8.559961584009904597206884504052, 9.114605605103133903707326114776, 10.57534600791652591081054214677, 10.98962583940140801249009851793, 11.59787809052696885108933576767, 11.88744328713633877464966356143, 13.11144064655754888494314970898, 13.13549784945325282811832403309

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