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
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
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_1$ | \( ( 1 + p^{11} T )^{2} \) |
3 | \( 1 \) | ||
good | 5 | $D_{4}$ | \( 1 + 25248156 T - 86756036237194 p^{3} T^{2} + 25248156 p^{23} T^{3} + p^{46} T^{4} \) |
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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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