↑

Properties

Label 4-18e2-1.1-c43e2-0-1
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $44436.0$
Root an. cond. $14.5189$
Motivic weight $43$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.19e6·2-s + 1.31e13·4-s + 9.81e14·5-s − 1.62e18·7-s + 3.68e19·8-s + 4.11e21·10-s − 3.33e22·11-s + 3.15e23·13-s − 6.80e24·14-s + 9.67e25·16-s − 9.25e25·17-s − 3.22e27·19-s + 1.29e28·20-s − 1.39e29·22-s + 9.17e28·23-s − 4.85e29·25-s + 1.32e30·26-s − 2.13e31·28-s + 3.16e31·29-s − 3.05e31·31-s + 2.43e32·32-s − 3.88e32·34-s − 1.59e33·35-s + 8.05e33·37-s − 1.35e34·38-s + 3.61e34·40-s + 2.84e34·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.920·5-s − 1.09·7-s + 1.41·8-s + 1.30·10-s − 1.35·11-s + 0.354·13-s − 1.55·14-s + 5/4·16-s − 0.324·17-s − 1.03·19-s + 1.38·20-s − 1.92·22-s + 0.484·23-s − 0.426·25-s + 0.500·26-s − 1.64·28-s + 1.14·29-s − 0.263·31-s + 1.06·32-s − 0.459·34-s − 1.00·35-s + 1.54·37-s − 1.46·38-s + 1.30·40-s + 0.601·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(44436.0\)
Root analytic conductor: \(14.5189\)
Motivic weight: \(43\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :43/2, 43/2),\ 1)\)

Particular Values

\(L(22)\) \(\approx\) \(12.23108500\)
\(L(\frac12)\) \(\approx\) \(12.23108500\)
\(L(\frac{45}{2})\) 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$C_1$ \( ( 1 - p^{21} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 196235007955284 p T + \)\(92\!\cdots\!66\)\( p^{6} T^{2} - 196235007955284 p^{44} T^{3} + p^{86} T^{4} \)
7$D_{4}$ \( 1 + 33089483181835184 p^{2} T + \)\(15\!\cdots\!50\)\( p^{6} T^{2} + 33089483181835184 p^{45} T^{3} + p^{86} T^{4} \)
11$D_{4}$ \( 1 + \)\(30\!\cdots\!60\)\( p T + \)\(59\!\cdots\!82\)\( p^{4} T^{2} + \)\(30\!\cdots\!60\)\( p^{44} T^{3} + p^{86} T^{4} \)
13$D_{4}$ \( 1 - \)\(31\!\cdots\!12\)\( T + \)\(91\!\cdots\!70\)\( p^{2} T^{2} - \)\(31\!\cdots\!12\)\( p^{43} T^{3} + p^{86} T^{4} \)
17$D_{4}$ \( 1 + \)\(92\!\cdots\!64\)\( T + \)\(19\!\cdots\!50\)\( p T^{2} + \)\(92\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \)
19$D_{4}$ \( 1 + \)\(16\!\cdots\!08\)\( p T + \)\(29\!\cdots\!66\)\( p^{3} T^{2} + \)\(16\!\cdots\!08\)\( p^{44} T^{3} + p^{86} T^{4} \)
23$D_{4}$ \( 1 - \)\(91\!\cdots\!00\)\( T + \)\(25\!\cdots\!58\)\( p T^{2} - \)\(91\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \)
29$D_{4}$ \( 1 - \)\(31\!\cdots\!28\)\( T + \)\(61\!\cdots\!06\)\( p T^{2} - \)\(31\!\cdots\!28\)\( p^{43} T^{3} + p^{86} T^{4} \)
31$D_{4}$ \( 1 + \)\(98\!\cdots\!04\)\( p T + \)\(25\!\cdots\!66\)\( p^{2} T^{2} + \)\(98\!\cdots\!04\)\( p^{44} T^{3} + p^{86} T^{4} \)
37$D_{4}$ \( 1 - \)\(80\!\cdots\!88\)\( T + \)\(17\!\cdots\!66\)\( p T^{2} - \)\(80\!\cdots\!88\)\( p^{43} T^{3} + p^{86} T^{4} \)
41$D_{4}$ \( 1 - \)\(28\!\cdots\!72\)\( T + \)\(35\!\cdots\!18\)\( p T^{2} - \)\(28\!\cdots\!72\)\( p^{43} T^{3} + p^{86} T^{4} \)
43$D_{4}$ \( 1 - \)\(14\!\cdots\!96\)\( T + \)\(25\!\cdots\!18\)\( T^{2} - \)\(14\!\cdots\!96\)\( p^{43} T^{3} + p^{86} T^{4} \)
47$D_{4}$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(77\!\cdots\!46\)\( T^{2} - \)\(11\!\cdots\!40\)\( p^{43} T^{3} + p^{86} T^{4} \)
53$D_{4}$ \( 1 - \)\(29\!\cdots\!76\)\( T + \)\(49\!\cdots\!98\)\( T^{2} - \)\(29\!\cdots\!76\)\( p^{43} T^{3} + p^{86} T^{4} \)
59$D_{4}$ \( 1 - \)\(27\!\cdots\!60\)\( T + \)\(27\!\cdots\!58\)\( T^{2} - \)\(27\!\cdots\!60\)\( p^{43} T^{3} + p^{86} T^{4} \)
61$D_{4}$ \( 1 - \)\(38\!\cdots\!60\)\( T + \)\(14\!\cdots\!62\)\( T^{2} - \)\(38\!\cdots\!60\)\( p^{43} T^{3} + p^{86} T^{4} \)
67$D_{4}$ \( 1 - \)\(98\!\cdots\!84\)\( T + \)\(67\!\cdots\!90\)\( T^{2} - \)\(98\!\cdots\!84\)\( p^{43} T^{3} + p^{86} T^{4} \)
71$D_{4}$ \( 1 - \)\(97\!\cdots\!20\)\( T + \)\(10\!\cdots\!22\)\( T^{2} - \)\(97\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
73$D_{4}$ \( 1 + \)\(32\!\cdots\!08\)\( T + \)\(52\!\cdots\!50\)\( T^{2} + \)\(32\!\cdots\!08\)\( p^{43} T^{3} + p^{86} T^{4} \)
79$D_{4}$ \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(11\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
83$D_{4}$ \( 1 - \)\(79\!\cdots\!08\)\( T + \)\(29\!\cdots\!90\)\( T^{2} - \)\(79\!\cdots\!08\)\( p^{43} T^{3} + p^{86} T^{4} \)
89$D_{4}$ \( 1 - \)\(80\!\cdots\!16\)\( T + \)\(12\!\cdots\!02\)\( T^{2} - \)\(80\!\cdots\!16\)\( p^{43} T^{3} + p^{86} T^{4} \)
97$D_{4}$ \( 1 - \)\(47\!\cdots\!88\)\( T + \)\(36\!\cdots\!82\)\( T^{2} - \)\(47\!\cdots\!88\)\( p^{43} T^{3} + p^{86} T^{4} \)
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

−11.29584726707280920876207101458, −10.78224377122135982347769808115, −10.06214828964953444536116562104, −10.01490514267390754446532417252, −8.953530416912852464146095361279, −8.477637205703269282874755423188, −7.51135520763882695733843769100, −7.15393431972912801738683559337, −6.30958735119087481584582273173, −6.08922382910006712053534303028, −5.58195439389897517773846074158, −5.09158313894833702990057972229, −4.17452979764782647810917026665, −4.09327828481911942737266340192, −3.12475841388329498681582950718, −2.65733171435044728476692695271, −2.34677321294535620288758271049, −1.87519101627653941236571887329, −0.75561427713294426384053522784, −0.61164522061743782568531071256, 0.61164522061743782568531071256, 0.75561427713294426384053522784, 1.87519101627653941236571887329, 2.34677321294535620288758271049, 2.65733171435044728476692695271, 3.12475841388329498681582950718, 4.09327828481911942737266340192, 4.17452979764782647810917026665, 5.09158313894833702990057972229, 5.58195439389897517773846074158, 6.08922382910006712053534303028, 6.30958735119087481584582273173, 7.15393431972912801738683559337, 7.51135520763882695733843769100, 8.477637205703269282874755423188, 8.953530416912852464146095361279, 10.01490514267390754446532417252, 10.06214828964953444536116562104, 10.78224377122135982347769808115, 11.29584726707280920876207101458

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