Properties

Label 16-63e8-1.1-c4e8-0-0
Degree $16$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $3.23503\times 10^{6}$
Root an. cond. $2.55192$
Motivic weight $4$
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  + 4·4-s − 360·13-s + 227·16-s − 888·19-s + 1.20e3·25-s + 1.11e3·31-s + 4.28e3·37-s − 1.28e4·43-s + 1.37e3·49-s − 1.44e3·52-s + 5.97e3·61-s + 3.73e3·64-s + 840·67-s + 2.24e4·73-s − 3.55e3·76-s + 3.20e4·79-s − 7.55e3·97-s + 4.81e3·100-s + 4.95e4·103-s − 4.42e4·109-s + 6.68e4·121-s + 4.44e3·124-s + 127-s + 131-s + 137-s + 139-s + 1.71e4·148-s + ⋯
L(s)  = 1  + 1/4·4-s − 2.13·13-s + 0.886·16-s − 2.45·19-s + 1.92·25-s + 1.15·31-s + 3.12·37-s − 6.92·43-s + 4/7·49-s − 0.532·52-s + 1.60·61-s + 0.911·64-s + 0.187·67-s + 4.21·73-s − 0.614·76-s + 5.13·79-s − 0.802·97-s + 0.481·100-s + 4.67·103-s − 3.72·109-s + 4.56·121-s + 0.289·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.781·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.23503\times 10^{6}\)
Root analytic conductor: \(2.55192\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [2]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5235842470\)
\(L(\frac12)\) \(\approx\) \(0.5235842470\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - p^{3} T^{2} )^{4} \)
good2 \( 1 - p^{2} T^{2} - 211 T^{4} - 495 p^{2} T^{6} + 18281 p^{2} T^{8} - 495 p^{10} T^{10} - 211 p^{16} T^{12} - p^{26} T^{14} + p^{32} T^{16} \)
5 \( 1 - 1204 T^{2} + 1088792 T^{4} - 496610268 T^{6} + 297978977966 T^{8} - 496610268 p^{8} T^{10} + 1088792 p^{16} T^{12} - 1204 p^{24} T^{14} + p^{32} T^{16} \)
11 \( 1 - 66820 T^{2} + 2410062440 T^{4} - 57863283311052 T^{6} + 990271739905879886 T^{8} - 57863283311052 p^{8} T^{10} + 2410062440 p^{16} T^{12} - 66820 p^{24} T^{14} + p^{32} T^{16} \)
13 \( ( 1 + 180 T + 49544 T^{2} + 7274268 T^{3} + 1011901710 T^{4} + 7274268 p^{4} T^{5} + 49544 p^{8} T^{6} + 180 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
17 \( 1 - 399668 T^{2} + 81228875608 T^{4} - 10853258660506460 T^{6} + \)\(10\!\cdots\!82\)\( T^{8} - 10853258660506460 p^{8} T^{10} + 81228875608 p^{16} T^{12} - 399668 p^{24} T^{14} + p^{32} T^{16} \)
19 \( ( 1 + 444 T + 537500 T^{2} + 159333108 T^{3} + 104585670390 T^{4} + 159333108 p^{4} T^{5} + 537500 p^{8} T^{6} + 444 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
23 \( 1 - 1402436 T^{2} + 1003247250280 T^{4} - 468866569301757452 T^{6} + \)\(15\!\cdots\!50\)\( T^{8} - 468866569301757452 p^{8} T^{10} + 1003247250280 p^{16} T^{12} - 1402436 p^{24} T^{14} + p^{32} T^{16} \)
29 \( 1 - 1623568 T^{2} + 1154187781892 T^{4} - 313003174452068400 T^{6} + \)\(57\!\cdots\!26\)\( T^{8} - 313003174452068400 p^{8} T^{10} + 1154187781892 p^{16} T^{12} - 1623568 p^{24} T^{14} + p^{32} T^{16} \)
31 \( ( 1 - 556 T + 828092 T^{2} - 399081348 T^{3} - 240163837450 T^{4} - 399081348 p^{4} T^{5} + 828092 p^{8} T^{6} - 556 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
37 \( ( 1 - 2140 T + 6320792 T^{2} - 10980125844 T^{3} + 17004933841070 T^{4} - 10980125844 p^{4} T^{5} + 6320792 p^{8} T^{6} - 2140 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
41 \( 1 - 15843412 T^{2} + 125515057580312 T^{4} - \)\(62\!\cdots\!12\)\( T^{6} + \)\(21\!\cdots\!98\)\( T^{8} - \)\(62\!\cdots\!12\)\( p^{8} T^{10} + 125515057580312 p^{16} T^{12} - 15843412 p^{24} T^{14} + p^{32} T^{16} \)
43 \( ( 1 + 6400 T + 27902068 T^{2} + 78411451264 T^{3} + 171378556550566 T^{4} + 78411451264 p^{4} T^{5} + 27902068 p^{8} T^{6} + 6400 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
47 \( 1 - 8901656 T^{2} + 24616537945276 T^{4} - \)\(18\!\cdots\!92\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(18\!\cdots\!92\)\( p^{8} T^{10} + 24616537945276 p^{16} T^{12} - 8901656 p^{24} T^{14} + p^{32} T^{16} \)
53 \( 1 - 30287264 T^{2} + 499842379923076 T^{4} - \)\(57\!\cdots\!92\)\( T^{6} + \)\(50\!\cdots\!22\)\( T^{8} - \)\(57\!\cdots\!92\)\( p^{8} T^{10} + 499842379923076 p^{16} T^{12} - 30287264 p^{24} T^{14} + p^{32} T^{16} \)
59 \( 1 - 15717656 T^{2} + 613397835693628 T^{4} - \)\(69\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!38\)\( T^{8} - \)\(69\!\cdots\!24\)\( p^{8} T^{10} + 613397835693628 p^{16} T^{12} - 15717656 p^{24} T^{14} + p^{32} T^{16} \)
61 \( ( 1 - 2988 T + 30076028 T^{2} - 35349881652 T^{3} + 433315161025926 T^{4} - 35349881652 p^{4} T^{5} + 30076028 p^{8} T^{6} - 2988 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
67 \( ( 1 - 420 T + 980360 p T^{2} - 6221257644 T^{3} + 1829179676189934 T^{4} - 6221257644 p^{4} T^{5} + 980360 p^{9} T^{6} - 420 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
71 \( 1 - 148460804 T^{2} + 10664159703683176 T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!42\)\( T^{8} - \)\(47\!\cdots\!52\)\( p^{8} T^{10} + 10664159703683176 p^{16} T^{12} - 148460804 p^{24} T^{14} + p^{32} T^{16} \)
73 \( ( 1 - 11224 T + 146430920 T^{2} - 966134182728 T^{3} + 6689580927026510 T^{4} - 966134182728 p^{4} T^{5} + 146430920 p^{8} T^{6} - 11224 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
79 \( ( 1 - 16028 T + 210444568 T^{2} - 1937189219636 T^{3} + 13381577795224942 T^{4} - 1937189219636 p^{4} T^{5} + 210444568 p^{8} T^{6} - 16028 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
83 \( 1 - 179839496 T^{2} + 16808804665405852 T^{4} - \)\(11\!\cdots\!48\)\( T^{6} + \)\(59\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!48\)\( p^{8} T^{10} + 16808804665405852 p^{16} T^{12} - 179839496 p^{24} T^{14} + p^{32} T^{16} \)
89 \( 1 - 379725172 T^{2} + 66612485799234008 T^{4} - \)\(72\!\cdots\!84\)\( T^{6} + \)\(53\!\cdots\!98\)\( T^{8} - \)\(72\!\cdots\!84\)\( p^{8} T^{10} + 66612485799234008 p^{16} T^{12} - 379725172 p^{24} T^{14} + p^{32} T^{16} \)
97 \( ( 1 + 3776 T + 241007096 T^{2} + 1051824414720 T^{3} + 28620303313344302 T^{4} + 1051824414720 p^{4} T^{5} + 241007096 p^{8} T^{6} + 3776 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.36107424915128519714369723834, −6.28050312977716951033920050910, −6.19274619693995616134055849746, −5.52372519080405722139834871345, −5.41097904714836616046094142232, −5.37065724748808645204859320521, −5.04350931075301314297425945463, −5.03663655322558579737330681604, −4.70670288996095455876544536667, −4.57808462726166095159995562828, −4.37362581139271143826126390713, −4.29062663599101510026673884687, −3.68473666430359316921141158353, −3.46958713373001705372204931759, −3.38533209013297939234642990971, −3.33203293074976775217160948748, −2.81914397574691855745188764783, −2.36862683764144697062727029054, −2.25671026035627156686196669711, −2.12112713368917280588071811498, −2.06213770613825694138936027118, −1.17598764156078513631807055260, −1.01512948052103528925864585656, −0.67273830345945414088742083203, −0.10175034889209775993302197032, 0.10175034889209775993302197032, 0.67273830345945414088742083203, 1.01512948052103528925864585656, 1.17598764156078513631807055260, 2.06213770613825694138936027118, 2.12112713368917280588071811498, 2.25671026035627156686196669711, 2.36862683764144697062727029054, 2.81914397574691855745188764783, 3.33203293074976775217160948748, 3.38533209013297939234642990971, 3.46958713373001705372204931759, 3.68473666430359316921141158353, 4.29062663599101510026673884687, 4.37362581139271143826126390713, 4.57808462726166095159995562828, 4.70670288996095455876544536667, 5.03663655322558579737330681604, 5.04350931075301314297425945463, 5.37065724748808645204859320521, 5.41097904714836616046094142232, 5.52372519080405722139834871345, 6.19274619693995616134055849746, 6.28050312977716951033920050910, 6.36107424915128519714369723834

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

Plot not available for L-functions of degree greater than 10.