Properties

Label 4-18e2-1.1-c23e2-0-1
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $3640.52$
Root an. cond. $7.76767$
Motivic weight $23$
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  − 4.09e3·2-s + 1.25e7·4-s − 2.52e7·5-s + 5.76e9·7-s − 3.43e10·8-s + 1.03e11·10-s − 1.01e12·11-s + 2.75e12·13-s − 2.36e13·14-s + 8.79e13·16-s − 5.83e13·17-s − 4.18e14·19-s − 3.17e14·20-s + 4.16e15·22-s − 1.50e16·23-s + 1.14e16·25-s − 1.12e16·26-s + 7.25e16·28-s + 6.98e16·29-s − 1.98e15·31-s − 2.16e17·32-s + 2.38e17·34-s − 1.45e17·35-s − 1.59e18·37-s + 1.71e18·38-s + 8.67e17·40-s + 8.08e18·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.231·5-s + 1.10·7-s − 1.41·8-s + 0.327·10-s − 1.07·11-s + 0.426·13-s − 1.55·14-s + 5/4·16-s − 0.412·17-s − 0.823·19-s − 0.346·20-s + 1.52·22-s − 3.30·23-s + 0.963·25-s − 0.603·26-s + 1.65·28-s + 1.06·29-s − 0.0140·31-s − 1.06·32-s + 0.583·34-s − 0.254·35-s − 1.47·37-s + 1.16·38-s + 0.327·40-s + 2.29·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3640.52\)
Root analytic conductor: \(7.76767\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 324,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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$C_1$ \( ( 1 + p^{11} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 25248156 T - 86756036237194 p^{3} T^{2} + 25248156 p^{23} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 5764462768 T + 842166761537839758 p^{2} T^{2} - 5764462768 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 92465588184 p T + \)\(63\!\cdots\!86\)\( p^{2} T^{2} + 92465588184 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 2755261084876 T + \)\(59\!\cdots\!26\)\( p T^{2} - 2755261084876 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 3429693163044 p T + \)\(63\!\cdots\!18\)\( p^{2} T^{2} + 3429693163044 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 22017839503640 p T + \)\(13\!\cdots\!38\)\( p^{2} T^{2} + 22017839503640 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 15094773537744336 T + \)\(96\!\cdots\!58\)\( T^{2} + 15094773537744336 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 69804282346510740 T + \)\(43\!\cdots\!78\)\( T^{2} - 69804282346510740 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 1989488015768576 T + \)\(13\!\cdots\!26\)\( T^{2} + 1989488015768576 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 1596195400758645092 T + \)\(25\!\cdots\!22\)\( T^{2} + 1596195400758645092 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 8085128920513043916 T + \)\(34\!\cdots\!06\)\( T^{2} - 8085128920513043916 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 4586671849073581304 T + \)\(77\!\cdots\!18\)\( T^{2} + 4586671849073581304 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 5860314694375864608 T + \)\(48\!\cdots\!62\)\( T^{2} + 5860314694375864608 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + 57187991382310436796 T + \)\(92\!\cdots\!58\)\( T^{2} + 57187991382310436796 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(17\!\cdots\!20\)\( T + \)\(45\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(69\!\cdots\!76\)\( T + \)\(31\!\cdots\!06\)\( T^{2} + \)\(69\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(89\!\cdots\!32\)\( T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(89\!\cdots\!32\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 + \)\(20\!\cdots\!84\)\( T + \)\(57\!\cdots\!86\)\( T^{2} + \)\(20\!\cdots\!84\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 - \)\(54\!\cdots\!36\)\( T + \)\(13\!\cdots\!58\)\( T^{2} - \)\(54\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + \)\(82\!\cdots\!40\)\( T + \)\(99\!\cdots\!78\)\( T^{2} + \)\(82\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(84\!\cdots\!84\)\( T + \)\(22\!\cdots\!38\)\( T^{2} - \)\(84\!\cdots\!84\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(14\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(29\!\cdots\!08\)\( T + \)\(10\!\cdots\!62\)\( T^{2} - \)\(29\!\cdots\!08\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−12.91905491369026376291972862911, −12.26655885487500542467414112921, −11.59025509505701495429784763562, −11.02821236518895556861312887697, −10.23399994363524666735890202245, −10.19023577045819708113018311405, −8.847891194632294191551148065776, −8.559869934178177927980164023336, −7.72642461350994235942254997101, −7.67523774394259125402493822267, −6.38765041571325572314130443925, −5.98702194845587304527710369447, −4.88155703355335459720449080218, −4.22626362473077770115041678181, −3.16228779871271828879966495228, −2.25845730301580369761351376427, −1.84646808673943561653134518078, −1.11741186235029363473254306730, 0, 0, 1.11741186235029363473254306730, 1.84646808673943561653134518078, 2.25845730301580369761351376427, 3.16228779871271828879966495228, 4.22626362473077770115041678181, 4.88155703355335459720449080218, 5.98702194845587304527710369447, 6.38765041571325572314130443925, 7.67523774394259125402493822267, 7.72642461350994235942254997101, 8.559869934178177927980164023336, 8.847891194632294191551148065776, 10.19023577045819708113018311405, 10.23399994363524666735890202245, 11.02821236518895556861312887697, 11.59025509505701495429784763562, 12.26655885487500542467414112921, 12.91905491369026376291972862911

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