Properties

Label 16-17e8-1.1-c3e8-0-0
Degree $16$
Conductor $6975757441$
Sign $1$
Analytic cond. $1.02451$
Root an. cond. $1.00151$
Motivic weight $3$
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  + 14·4-s + 14·5-s + 2·7-s − 108·11-s − 88·13-s + 131·16-s − 10·17-s + 196·20-s − 22·23-s + 98·25-s + 180·27-s + 28·28-s + 46·29-s + 610·31-s + 28·35-s − 574·37-s − 968·41-s − 1.51e3·44-s − 368·47-s + 2·49-s − 1.23e3·52-s − 1.51e3·55-s + 1.25e3·61-s + 972·64-s − 1.23e3·65-s + 764·67-s − 140·68-s + ⋯
L(s)  = 1  + 7/4·4-s + 1.25·5-s + 0.107·7-s − 2.96·11-s − 1.87·13-s + 2.04·16-s − 0.142·17-s + 2.19·20-s − 0.199·23-s + 0.783·25-s + 1.28·27-s + 0.188·28-s + 0.294·29-s + 3.53·31-s + 0.135·35-s − 2.55·37-s − 3.68·41-s − 5.18·44-s − 1.14·47-s + 0.00583·49-s − 3.28·52-s − 3.70·55-s + 2.64·61-s + 1.89·64-s − 2.35·65-s + 1.39·67-s − 0.249·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(17^{8}\)
Sign: $1$
Analytic conductor: \(1.02451\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 17^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.592863866\)
\(L(\frac12)\) \(\approx\) \(1.592863866\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + 10 T - 160 p T^{2} - 10 p^{2} T^{3} + 9246 p^{3} T^{4} - 10 p^{5} T^{5} - 160 p^{7} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
good2 \( 1 - 7 p T^{2} + 65 T^{4} - 3 p^{4} T^{6} - 9 p^{7} T^{8} - 3 p^{10} T^{10} + 65 p^{12} T^{12} - 7 p^{19} T^{14} + p^{24} T^{16} \)
3 \( 1 - 20 p^{2} T^{3} - 124 T^{4} + 500 p^{2} T^{5} + 200 p^{4} T^{6} + 200 p^{2} T^{7} - 880154 T^{8} + 200 p^{5} T^{9} + 200 p^{10} T^{10} + 500 p^{11} T^{11} - 124 p^{12} T^{12} - 20 p^{17} T^{13} + p^{24} T^{16} \)
5 \( 1 - 14 T + 98 T^{2} - 1742 T^{3} + 12064 T^{4} + 83238 T^{5} - 830322 T^{6} + 22258486 T^{7} - 523179234 T^{8} + 22258486 p^{3} T^{9} - 830322 p^{6} T^{10} + 83238 p^{9} T^{11} + 12064 p^{12} T^{12} - 1742 p^{15} T^{13} + 98 p^{18} T^{14} - 14 p^{21} T^{15} + p^{24} T^{16} \)
7 \( 1 - 2 T + 2 T^{2} - 10226 T^{3} + 81296 T^{4} + 194114 p T^{5} + 49405350 T^{6} - 812471938 T^{7} - 20043782946 T^{8} - 812471938 p^{3} T^{9} + 49405350 p^{6} T^{10} + 194114 p^{10} T^{11} + 81296 p^{12} T^{12} - 10226 p^{15} T^{13} + 2 p^{18} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} \)
11 \( 1 + 108 T + 5832 T^{2} + 285984 T^{3} + 1320492 p T^{4} + 631823112 T^{5} + 24418117440 T^{6} + 981926711892 T^{7} + 38344333256678 T^{8} + 981926711892 p^{3} T^{9} + 24418117440 p^{6} T^{10} + 631823112 p^{9} T^{11} + 1320492 p^{13} T^{12} + 285984 p^{15} T^{13} + 5832 p^{18} T^{14} + 108 p^{21} T^{15} + p^{24} T^{16} \)
13 \( ( 1 + 44 T + 6452 T^{2} + 185644 T^{3} + 18227830 T^{4} + 185644 p^{3} T^{5} + 6452 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
19 \( 1 - 49548 T^{2} + 1108248628 T^{4} - 14581622201236 T^{6} + 123306344393137910 T^{8} - 14581622201236 p^{6} T^{10} + 1108248628 p^{12} T^{12} - 49548 p^{18} T^{14} + p^{24} T^{16} \)
23 \( 1 + 22 T + 242 T^{2} + 305150 T^{3} + 283710864 T^{4} + 86720882 p T^{5} + 21780998454 T^{6} + 34099874325206 T^{7} + 50076554997930590 T^{8} + 34099874325206 p^{3} T^{9} + 21780998454 p^{6} T^{10} + 86720882 p^{10} T^{11} + 283710864 p^{12} T^{12} + 305150 p^{15} T^{13} + 242 p^{18} T^{14} + 22 p^{21} T^{15} + p^{24} T^{16} \)
29 \( 1 - 46 T + 1058 T^{2} - 350894 T^{3} - 1469341632 T^{4} + 48677544854 T^{5} - 623040317010 T^{6} - 326680376209530 T^{7} + 1059567541310947486 T^{8} - 326680376209530 p^{3} T^{9} - 623040317010 p^{6} T^{10} + 48677544854 p^{9} T^{11} - 1469341632 p^{12} T^{12} - 350894 p^{15} T^{13} + 1058 p^{18} T^{14} - 46 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 610 T + 186050 T^{2} - 45370570 T^{3} + 7376742336 T^{4} - 319696487050 T^{5} - 148183743449850 T^{6} + 60089442929313630 T^{7} - 13665887910277786754 T^{8} + 60089442929313630 p^{3} T^{9} - 148183743449850 p^{6} T^{10} - 319696487050 p^{9} T^{11} + 7376742336 p^{12} T^{12} - 45370570 p^{15} T^{13} + 186050 p^{18} T^{14} - 610 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 + 574 T + 164738 T^{2} + 36289990 T^{3} + 6240461504 T^{4} + 1192660509522 T^{5} + 315027672319726 T^{6} + 82684080067897738 T^{7} + 20284670290250127070 T^{8} + 82684080067897738 p^{3} T^{9} + 315027672319726 p^{6} T^{10} + 1192660509522 p^{9} T^{11} + 6240461504 p^{12} T^{12} + 36289990 p^{15} T^{13} + 164738 p^{18} T^{14} + 574 p^{21} T^{15} + p^{24} T^{16} \)
41 \( 1 + 968 T + 468512 T^{2} + 198428824 T^{3} + 85385967612 T^{4} + 748503298312 p T^{5} + 9389247542584800 T^{6} + 2850342553409293272 T^{7} + \)\(81\!\cdots\!98\)\( T^{8} + 2850342553409293272 p^{3} T^{9} + 9389247542584800 p^{6} T^{10} + 748503298312 p^{10} T^{11} + 85385967612 p^{12} T^{12} + 198428824 p^{15} T^{13} + 468512 p^{18} T^{14} + 968 p^{21} T^{15} + p^{24} T^{16} \)
43 \( 1 - 524936 T^{2} + 2966357780 p T^{4} - 18699551386799032 T^{6} + \)\(18\!\cdots\!98\)\( T^{8} - 18699551386799032 p^{6} T^{10} + 2966357780 p^{13} T^{12} - 524936 p^{18} T^{14} + p^{24} T^{16} \)
47 \( ( 1 + 184 T + 303628 T^{2} + 45624920 T^{3} + 43219349926 T^{4} + 45624920 p^{3} T^{5} + 303628 p^{6} T^{6} + 184 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
53 \( 1 - 429476 T^{2} + 132292785860 T^{4} - 29887976589939132 T^{6} + \)\(48\!\cdots\!18\)\( T^{8} - 29887976589939132 p^{6} T^{10} + 132292785860 p^{12} T^{12} - 429476 p^{18} T^{14} + p^{24} T^{16} \)
59 \( 1 - 803128 T^{2} + 306274528508 T^{4} - 82679168899041416 T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - 82679168899041416 p^{6} T^{10} + 306274528508 p^{12} T^{12} - 803128 p^{18} T^{14} + p^{24} T^{16} \)
61 \( 1 - 1258 T + 791282 T^{2} - 465239298 T^{3} + 289545560288 T^{4} - 163355761361878 T^{5} + 84613159959199710 T^{6} - 47961410201311305310 T^{7} + \)\(25\!\cdots\!66\)\( T^{8} - 47961410201311305310 p^{3} T^{9} + 84613159959199710 p^{6} T^{10} - 163355761361878 p^{9} T^{11} + 289545560288 p^{12} T^{12} - 465239298 p^{15} T^{13} + 791282 p^{18} T^{14} - 1258 p^{21} T^{15} + p^{24} T^{16} \)
67 \( ( 1 - 382 T + 518844 T^{2} + 127863586 T^{3} + 50809502070 T^{4} + 127863586 p^{3} T^{5} + 518844 p^{6} T^{6} - 382 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( 1 - 1266 T + 801378 T^{2} - 406667714 T^{3} + 257625327744 T^{4} - 188037640245138 T^{5} + 114289697458506374 T^{6} - 42774297641864818786 T^{7} + \)\(12\!\cdots\!34\)\( T^{8} - 42774297641864818786 p^{3} T^{9} + 114289697458506374 p^{6} T^{10} - 188037640245138 p^{9} T^{11} + 257625327744 p^{12} T^{12} - 406667714 p^{15} T^{13} + 801378 p^{18} T^{14} - 1266 p^{21} T^{15} + p^{24} T^{16} \)
73 \( 1 + 1732 T + 1499912 T^{2} + 1036616588 T^{3} + 594980634828 T^{4} + 355861503522212 T^{5} + 261220505412716920 T^{6} + \)\(20\!\cdots\!24\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} + \)\(20\!\cdots\!24\)\( p^{3} T^{9} + 261220505412716920 p^{6} T^{10} + 355861503522212 p^{9} T^{11} + 594980634828 p^{12} T^{12} + 1036616588 p^{15} T^{13} + 1499912 p^{18} T^{14} + 1732 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 - 914 T + 417698 T^{2} + 360102614 T^{3} + 55218179088 T^{4} - 435936491163994 T^{5} + 440218376260007590 T^{6} - 25818998319394847330 T^{7} - \)\(81\!\cdots\!34\)\( T^{8} - 25818998319394847330 p^{3} T^{9} + 440218376260007590 p^{6} T^{10} - 435936491163994 p^{9} T^{11} + 55218179088 p^{12} T^{12} + 360102614 p^{15} T^{13} + 417698 p^{18} T^{14} - 914 p^{21} T^{15} + p^{24} T^{16} \)
83 \( 1 - 2421080 T^{2} + 2978675688956 T^{4} - 2441080253881110760 T^{6} + \)\(15\!\cdots\!06\)\( T^{8} - 2441080253881110760 p^{6} T^{10} + 2978675688956 p^{12} T^{12} - 2421080 p^{18} T^{14} + p^{24} T^{16} \)
89 \( ( 1 + 1078 T + 2962392 T^{2} + 2155154546 T^{3} + 3159947923854 T^{4} + 2155154546 p^{3} T^{5} + 2962392 p^{6} T^{6} + 1078 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
97 \( 1 - 1836 T + 1685448 T^{2} - 1048047300 T^{3} + 1167661514444 T^{4} - 1645617931539948 T^{5} + 1602523329629378616 T^{6} - \)\(45\!\cdots\!92\)\( T^{7} - \)\(36\!\cdots\!30\)\( T^{8} - \)\(45\!\cdots\!92\)\( p^{3} T^{9} + 1602523329629378616 p^{6} T^{10} - 1645617931539948 p^{9} T^{11} + 1167661514444 p^{12} T^{12} - 1048047300 p^{15} T^{13} + 1685448 p^{18} T^{14} - 1836 p^{21} T^{15} + p^{24} T^{16} \)
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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

−9.354151487172606028043634384289, −8.760918225907517778717444941318, −8.697789739397150969722747861959, −8.419785836937399860036460283773, −8.202857578659883471533014340052, −8.199066667541582882068144768172, −8.008786003122701774754681558182, −7.51881575581137225411035006800, −7.16939616777459365669093710526, −6.99890415722263368711418769673, −6.90608271651863634775251382964, −6.90363798133140020169176602221, −6.20715737268211484634574494125, −6.07904238286820764077112287950, −6.04820426604958289261007283157, −5.17691914227545174080496313913, −5.17228505607946442845161025837, −5.07866348333236228723257195587, −4.96089457649031419736760280302, −4.41212984817514497884039459901, −3.36869997447903686472112275929, −3.21084383272816080884314557330, −2.67027849164515275936211708830, −2.35290567019488074780878588608, −1.98982379708749078810849432379, 1.98982379708749078810849432379, 2.35290567019488074780878588608, 2.67027849164515275936211708830, 3.21084383272816080884314557330, 3.36869997447903686472112275929, 4.41212984817514497884039459901, 4.96089457649031419736760280302, 5.07866348333236228723257195587, 5.17228505607946442845161025837, 5.17691914227545174080496313913, 6.04820426604958289261007283157, 6.07904238286820764077112287950, 6.20715737268211484634574494125, 6.90363798133140020169176602221, 6.90608271651863634775251382964, 6.99890415722263368711418769673, 7.16939616777459365669093710526, 7.51881575581137225411035006800, 8.008786003122701774754681558182, 8.199066667541582882068144768172, 8.202857578659883471533014340052, 8.419785836937399860036460283773, 8.697789739397150969722747861959, 8.760918225907517778717444941318, 9.354151487172606028043634384289

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

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