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Properties

Label 4-18e2-1.1-c41e2-0-1
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $36729.3$
Root an. cond. $13.8437$
Motivic weight $41$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09e6·2-s + 3.29e12·4-s − 9.75e13·5-s + 2.17e17·7-s − 4.61e18·8-s + 2.04e20·10-s − 1.10e20·11-s + 1.73e23·13-s − 4.55e23·14-s + 6.04e24·16-s + 1.02e25·17-s + 2.06e26·19-s − 3.21e26·20-s + 2.31e26·22-s − 1.49e28·23-s + 8.05e27·25-s − 3.63e29·26-s + 7.16e29·28-s + 1.37e30·29-s − 3.62e30·31-s − 7.60e30·32-s − 2.15e31·34-s − 2.11e31·35-s + 1.20e32·37-s − 4.32e32·38-s + 4.50e32·40-s − 1.27e33·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.457·5-s + 1.02·7-s − 1.41·8-s + 0.647·10-s − 0.0495·11-s + 2.52·13-s − 1.45·14-s + 5/4·16-s + 0.613·17-s + 1.25·19-s − 0.686·20-s + 0.0700·22-s − 1.81·23-s + 0.177·25-s − 3.57·26-s + 1.54·28-s + 1.44·29-s − 0.969·31-s − 1.06·32-s − 0.868·34-s − 0.470·35-s + 0.855·37-s − 1.78·38-s + 0.647·40-s − 1.10·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(36729.3\)
Root analytic conductor: \(13.8437\)
Motivic weight: \(41\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :41/2, 41/2),\ 1)\)

Particular Values

\(L(21)\) \(\approx\) \(1.761443990\)
\(L(\frac12)\) \(\approx\) \(1.761443990\)
\(L(\frac{43}{2})\) 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$C_1$ \( ( 1 + p^{20} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 19519836865116 p T + \)\(47\!\cdots\!46\)\( p^{5} T^{2} + 19519836865116 p^{42} T^{3} + p^{82} T^{4} \)
7$D_{4}$ \( 1 - 4430751068838544 p^{2} T + \)\(79\!\cdots\!86\)\( p^{7} T^{2} - 4430751068838544 p^{43} T^{3} + p^{82} T^{4} \)
11$D_{4}$ \( 1 + 10054043771358142344 p T + \)\(64\!\cdots\!06\)\( p^{3} T^{2} + 10054043771358142344 p^{42} T^{3} + p^{82} T^{4} \)
13$D_{4}$ \( 1 - \)\(13\!\cdots\!36\)\( p T + \)\(76\!\cdots\!06\)\( p^{3} T^{2} - \)\(13\!\cdots\!36\)\( p^{42} T^{3} + p^{82} T^{4} \)
17$D_{4}$ \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(34\!\cdots\!14\)\( p T^{2} - \)\(10\!\cdots\!04\)\( p^{41} T^{3} + p^{82} T^{4} \)
19$D_{4}$ \( 1 - \)\(10\!\cdots\!00\)\( p T + \)\(65\!\cdots\!82\)\( p^{3} T^{2} - \)\(10\!\cdots\!00\)\( p^{42} T^{3} + p^{82} T^{4} \)
23$D_{4}$ \( 1 + \)\(14\!\cdots\!68\)\( T + \)\(82\!\cdots\!74\)\( p T^{2} + \)\(14\!\cdots\!68\)\( p^{41} T^{3} + p^{82} T^{4} \)
29$D_{4}$ \( 1 - \)\(47\!\cdots\!80\)\( p T + \)\(47\!\cdots\!02\)\( p T^{2} - \)\(47\!\cdots\!80\)\( p^{42} T^{3} + p^{82} T^{4} \)
31$D_{4}$ \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(10\!\cdots\!66\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{42} T^{3} + p^{82} T^{4} \)
37$D_{4}$ \( 1 - \)\(32\!\cdots\!28\)\( p T + \)\(31\!\cdots\!42\)\( p^{2} T^{2} - \)\(32\!\cdots\!28\)\( p^{42} T^{3} + p^{82} T^{4} \)
41$D_{4}$ \( 1 + \)\(12\!\cdots\!64\)\( T + \)\(75\!\cdots\!06\)\( T^{2} + \)\(12\!\cdots\!64\)\( p^{41} T^{3} + p^{82} T^{4} \)
43$D_{4}$ \( 1 + \)\(60\!\cdots\!52\)\( T + \)\(20\!\cdots\!62\)\( T^{2} + \)\(60\!\cdots\!52\)\( p^{41} T^{3} + p^{82} T^{4} \)
47$D_{4}$ \( 1 - \)\(25\!\cdots\!64\)\( T + \)\(84\!\cdots\!18\)\( T^{2} - \)\(25\!\cdots\!64\)\( p^{41} T^{3} + p^{82} T^{4} \)
53$D_{4}$ \( 1 - \)\(30\!\cdots\!52\)\( T + \)\(11\!\cdots\!82\)\( T^{2} - \)\(30\!\cdots\!52\)\( p^{41} T^{3} + p^{82} T^{4} \)
59$D_{4}$ \( 1 - \)\(32\!\cdots\!40\)\( T + \)\(78\!\cdots\!18\)\( T^{2} - \)\(32\!\cdots\!40\)\( p^{41} T^{3} + p^{82} T^{4} \)
61$D_{4}$ \( 1 + \)\(44\!\cdots\!36\)\( T + \)\(17\!\cdots\!46\)\( T^{2} + \)\(44\!\cdots\!36\)\( p^{41} T^{3} + p^{82} T^{4} \)
67$D_{4}$ \( 1 - \)\(55\!\cdots\!56\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(55\!\cdots\!56\)\( p^{41} T^{3} + p^{82} T^{4} \)
71$D_{4}$ \( 1 - \)\(48\!\cdots\!16\)\( T + \)\(10\!\cdots\!06\)\( T^{2} - \)\(48\!\cdots\!16\)\( p^{41} T^{3} + p^{82} T^{4} \)
73$D_{4}$ \( 1 + \)\(48\!\cdots\!12\)\( T + \)\(10\!\cdots\!82\)\( T^{2} + \)\(48\!\cdots\!12\)\( p^{41} T^{3} + p^{82} T^{4} \)
79$D_{4}$ \( 1 + \)\(90\!\cdots\!80\)\( T + \)\(14\!\cdots\!58\)\( T^{2} + \)\(90\!\cdots\!80\)\( p^{41} T^{3} + p^{82} T^{4} \)
83$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( T + \)\(95\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!68\)\( p^{41} T^{3} + p^{82} T^{4} \)
89$D_{4}$ \( 1 + \)\(26\!\cdots\!80\)\( T + \)\(16\!\cdots\!78\)\( T^{2} + \)\(26\!\cdots\!80\)\( p^{41} T^{3} + p^{82} T^{4} \)
97$D_{4}$ \( 1 + \)\(31\!\cdots\!04\)\( T + \)\(59\!\cdots\!98\)\( T^{2} + \)\(31\!\cdots\!04\)\( p^{41} T^{3} + p^{82} T^{4} \)
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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

−11.24668537808562248109250169042, −10.94764940544950934328592798944, −10.02527888858539207362564263066, −10.00680913205531129915204826898, −8.805657808129777018787796632565, −8.618186728282304043285324441965, −7.994295502719634804464993986498, −7.85177207254758747636100552971, −6.91632066048364874930436996464, −6.48512300543138795703245963629, −5.60402119714019884012147841426, −5.44324454746111934595877879251, −4.24037386602703644512312675531, −3.80836550658418851833411147693, −3.20286254881578005646761105727, −2.55690663113229286641490756867, −1.55810837416052306086266868404, −1.49548189059772227387219824473, −0.974386826799063667320026464545, −0.35547967204985572960072681853, 0.35547967204985572960072681853, 0.974386826799063667320026464545, 1.49548189059772227387219824473, 1.55810837416052306086266868404, 2.55690663113229286641490756867, 3.20286254881578005646761105727, 3.80836550658418851833411147693, 4.24037386602703644512312675531, 5.44324454746111934595877879251, 5.60402119714019884012147841426, 6.48512300543138795703245963629, 6.91632066048364874930436996464, 7.85177207254758747636100552971, 7.994295502719634804464993986498, 8.618186728282304043285324441965, 8.805657808129777018787796632565, 10.00680913205531129915204826898, 10.02527888858539207362564263066, 10.94764940544950934328592798944, 11.24668537808562248109250169042

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