Properties

Label 16-177e8-1.1-c3e8-0-0
Degree $16$
Conductor $9.634\times 10^{17}$
Sign $1$
Analytic cond. $1.41486\times 10^{8}$
Root an. cond. $3.23161$
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  + 2·2-s − 24·3-s − 11·4-s − 12·5-s − 48·6-s + 53·7-s − 29·8-s + 324·9-s − 24·10-s − 27·11-s + 264·12-s + 89·13-s + 106·14-s + 288·15-s + 65·16-s + 79·17-s + 648·18-s + 288·19-s + 132·20-s − 1.27e3·21-s − 54·22-s + 202·23-s + 696·24-s − 296·25-s + 178·26-s − 3.24e3·27-s − 583·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 4.61·3-s − 1.37·4-s − 1.07·5-s − 3.26·6-s + 2.86·7-s − 1.28·8-s + 12·9-s − 0.758·10-s − 0.740·11-s + 6.35·12-s + 1.89·13-s + 2.02·14-s + 4.95·15-s + 1.01·16-s + 1.12·17-s + 8.48·18-s + 3.47·19-s + 1.47·20-s − 13.2·21-s − 0.523·22-s + 1.83·23-s + 5.91·24-s − 2.36·25-s + 1.34·26-s − 23.0·27-s − 3.93·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 59^{8}\)
Sign: $1$
Analytic conductor: \(1.41486\times 10^{8}\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 59^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.810886887\)
\(L(\frac12)\) \(\approx\) \(4.810886887\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p T )^{8} \)
59 \( ( 1 + p T )^{8} \)
good2 \( 1 - p T + 15 T^{2} - 23 T^{3} + 11 p^{3} T^{4} - 151 T^{5} + 223 p^{2} T^{6} - 373 p^{2} T^{7} + 153 p^{6} T^{8} - 373 p^{5} T^{9} + 223 p^{8} T^{10} - 151 p^{9} T^{11} + 11 p^{15} T^{12} - 23 p^{15} T^{13} + 15 p^{18} T^{14} - p^{22} T^{15} + p^{24} T^{16} \)
5 \( 1 + 12 T + 88 p T^{2} + 4784 T^{3} + 97777 T^{4} + 1226524 T^{5} + 15794422 T^{6} + 209853804 T^{7} + 2037870536 T^{8} + 209853804 p^{3} T^{9} + 15794422 p^{6} T^{10} + 1226524 p^{9} T^{11} + 97777 p^{12} T^{12} + 4784 p^{15} T^{13} + 88 p^{19} T^{14} + 12 p^{21} T^{15} + p^{24} T^{16} \)
7 \( 1 - 53 T + 2603 T^{2} - 81412 T^{3} + 2487861 T^{4} - 60875803 T^{5} + 1470654937 T^{6} - 4303144536 p T^{7} + 602390593548 T^{8} - 4303144536 p^{4} T^{9} + 1470654937 p^{6} T^{10} - 60875803 p^{9} T^{11} + 2487861 p^{12} T^{12} - 81412 p^{15} T^{13} + 2603 p^{18} T^{14} - 53 p^{21} T^{15} + p^{24} T^{16} \)
11 \( 1 + 27 T + 4706 T^{2} + 102661 T^{3} + 11194641 T^{4} + 24478996 p T^{5} + 20331162878 T^{6} + 536574654998 T^{7} + 30150319050716 T^{8} + 536574654998 p^{3} T^{9} + 20331162878 p^{6} T^{10} + 24478996 p^{10} T^{11} + 11194641 p^{12} T^{12} + 102661 p^{15} T^{13} + 4706 p^{18} T^{14} + 27 p^{21} T^{15} + p^{24} T^{16} \)
13 \( 1 - 89 T + 12332 T^{2} - 849051 T^{3} + 71456129 T^{4} - 4222086862 T^{5} + 267359700706 T^{6} - 13501472811236 T^{7} + 694454725138108 T^{8} - 13501472811236 p^{3} T^{9} + 267359700706 p^{6} T^{10} - 4222086862 p^{9} T^{11} + 71456129 p^{12} T^{12} - 849051 p^{15} T^{13} + 12332 p^{18} T^{14} - 89 p^{21} T^{15} + p^{24} T^{16} \)
17 \( 1 - 79 T + 28503 T^{2} - 1945384 T^{3} + 385217373 T^{4} - 22809199409 T^{5} + 3256991619599 T^{6} - 166158049224224 T^{7} + 19001788511766504 T^{8} - 166158049224224 p^{3} T^{9} + 3256991619599 p^{6} T^{10} - 22809199409 p^{9} T^{11} + 385217373 p^{12} T^{12} - 1945384 p^{15} T^{13} + 28503 p^{18} T^{14} - 79 p^{21} T^{15} + p^{24} T^{16} \)
19 \( 1 - 288 T + 58747 T^{2} - 8266066 T^{3} + 1010575073 T^{4} - 5596563214 p T^{5} + 10729383625021 T^{6} - 985979638577952 T^{7} + 86198809543733100 T^{8} - 985979638577952 p^{3} T^{9} + 10729383625021 p^{6} T^{10} - 5596563214 p^{10} T^{11} + 1010575073 p^{12} T^{12} - 8266066 p^{15} T^{13} + 58747 p^{18} T^{14} - 288 p^{21} T^{15} + p^{24} T^{16} \)
23 \( 1 - 202 T + 78877 T^{2} - 9782904 T^{3} + 2189022771 T^{4} - 164008707750 T^{5} + 30647231120903 T^{6} - 1301563016452928 T^{7} + 337875564580174096 T^{8} - 1301563016452928 p^{3} T^{9} + 30647231120903 p^{6} T^{10} - 164008707750 p^{9} T^{11} + 2189022771 p^{12} T^{12} - 9782904 p^{15} T^{13} + 78877 p^{18} T^{14} - 202 p^{21} T^{15} + p^{24} T^{16} \)
29 \( 1 + 114 T + 142407 T^{2} + 15100644 T^{3} + 323886049 p T^{4} + 922057358314 T^{5} + 385335404059951 T^{6} + 34097901872698400 T^{7} + 11045871823285885480 T^{8} + 34097901872698400 p^{3} T^{9} + 385335404059951 p^{6} T^{10} + 922057358314 p^{9} T^{11} + 323886049 p^{13} T^{12} + 15100644 p^{15} T^{13} + 142407 p^{18} T^{14} + 114 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 538 T + 247841 T^{2} - 71976680 T^{3} + 19977570141 T^{4} - 4430907831460 T^{5} + 1001149670655357 T^{6} - 190849027976008154 T^{7} + 36094474572416755176 T^{8} - 190849027976008154 p^{3} T^{9} + 1001149670655357 p^{6} T^{10} - 4430907831460 p^{9} T^{11} + 19977570141 p^{12} T^{12} - 71976680 p^{15} T^{13} + 247841 p^{18} T^{14} - 538 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 - 395 T + 193049 T^{2} - 73775986 T^{3} + 24434845437 T^{4} - 7077501473739 T^{5} + 2026736803030389 T^{6} - 495192157890126338 T^{7} + \)\(11\!\cdots\!44\)\( T^{8} - 495192157890126338 p^{3} T^{9} + 2026736803030389 p^{6} T^{10} - 7077501473739 p^{9} T^{11} + 24434845437 p^{12} T^{12} - 73775986 p^{15} T^{13} + 193049 p^{18} T^{14} - 395 p^{21} T^{15} + p^{24} T^{16} \)
41 \( 1 + 39 T + 208219 T^{2} + 6579732 T^{3} + 27221850377 T^{4} + 1003358403829 T^{5} + 2746244448481691 T^{6} + 85963613205896072 T^{7} + \)\(21\!\cdots\!32\)\( T^{8} + 85963613205896072 p^{3} T^{9} + 2746244448481691 p^{6} T^{10} + 1003358403829 p^{9} T^{11} + 27221850377 p^{12} T^{12} + 6579732 p^{15} T^{13} + 208219 p^{18} T^{14} + 39 p^{21} T^{15} + p^{24} T^{16} \)
43 \( 1 - 527 T + 290648 T^{2} - 74281045 T^{3} + 33478254321 T^{4} - 9134887323040 T^{5} + 4044711537823774 T^{6} - 916042201060083954 T^{7} + \)\(33\!\cdots\!00\)\( T^{8} - 916042201060083954 p^{3} T^{9} + 4044711537823774 p^{6} T^{10} - 9134887323040 p^{9} T^{11} + 33478254321 p^{12} T^{12} - 74281045 p^{15} T^{13} + 290648 p^{18} T^{14} - 527 p^{21} T^{15} + p^{24} T^{16} \)
47 \( 1 - 860 T + 617297 T^{2} - 255345688 T^{3} + 85136887087 T^{4} - 13380662663840 T^{5} - 2156337380009501 T^{6} + 2872299573213577724 T^{7} - \)\(11\!\cdots\!24\)\( T^{8} + 2872299573213577724 p^{3} T^{9} - 2156337380009501 p^{6} T^{10} - 13380662663840 p^{9} T^{11} + 85136887087 p^{12} T^{12} - 255345688 p^{15} T^{13} + 617297 p^{18} T^{14} - 860 p^{21} T^{15} + p^{24} T^{16} \)
53 \( 1 + 812 T + 1220244 T^{2} + 739104612 T^{3} + 627073491445 T^{4} + 300484254323316 T^{5} + 183696967229381698 T^{6} + 71133339709305235224 T^{7} + \)\(33\!\cdots\!20\)\( T^{8} + 71133339709305235224 p^{3} T^{9} + 183696967229381698 p^{6} T^{10} + 300484254323316 p^{9} T^{11} + 627073491445 p^{12} T^{12} + 739104612 p^{15} T^{13} + 1220244 p^{18} T^{14} + 812 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 + 460 T + 905863 T^{2} + 353430850 T^{3} + 443432857027 T^{4} + 142106678937976 T^{5} + 142158849164093667 T^{6} + 39228985937752918146 T^{7} + \)\(35\!\cdots\!72\)\( T^{8} + 39228985937752918146 p^{3} T^{9} + 142158849164093667 p^{6} T^{10} + 142106678937976 p^{9} T^{11} + 443432857027 p^{12} T^{12} + 353430850 p^{15} T^{13} + 905863 p^{18} T^{14} + 460 p^{21} T^{15} + p^{24} T^{16} \)
67 \( 1 - 1934 T + 3576358 T^{2} - 4236276170 T^{3} + 4595486578449 T^{4} - 3911796347093456 T^{5} + 3053092890912856798 T^{6} - \)\(19\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!80\)\( T^{8} - \)\(19\!\cdots\!16\)\( p^{3} T^{9} + 3053092890912856798 p^{6} T^{10} - 3911796347093456 p^{9} T^{11} + 4595486578449 p^{12} T^{12} - 4236276170 p^{15} T^{13} + 3576358 p^{18} T^{14} - 1934 p^{21} T^{15} + p^{24} T^{16} \)
71 \( 1 + 1687 T + 3195198 T^{2} + 3608791647 T^{3} + 4054410195177 T^{4} + 3462592642030110 T^{5} + 2869857900682206934 T^{6} + \)\(19\!\cdots\!48\)\( T^{7} + \)\(17\!\cdots\!20\)\( p T^{8} + \)\(19\!\cdots\!48\)\( p^{3} T^{9} + 2869857900682206934 p^{6} T^{10} + 3462592642030110 p^{9} T^{11} + 4054410195177 p^{12} T^{12} + 3608791647 p^{15} T^{13} + 3195198 p^{18} T^{14} + 1687 p^{21} T^{15} + p^{24} T^{16} \)
73 \( 1 - 1980 T + 3765701 T^{2} - 4700098806 T^{3} + 5489032516989 T^{4} - 5065861518964928 T^{5} + 4376045351899850723 T^{6} - 43312987380720565790 p T^{7} + \)\(21\!\cdots\!56\)\( T^{8} - 43312987380720565790 p^{4} T^{9} + 4376045351899850723 p^{6} T^{10} - 5065861518964928 p^{9} T^{11} + 5489032516989 p^{12} T^{12} - 4700098806 p^{15} T^{13} + 3765701 p^{18} T^{14} - 1980 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 - 3319 T + 7263714 T^{2} - 11058483081 T^{3} + 13842945944569 T^{4} - 14385104804172608 T^{5} + 13296340874080788262 T^{6} - \)\(10\!\cdots\!10\)\( T^{7} + \)\(80\!\cdots\!04\)\( T^{8} - \)\(10\!\cdots\!10\)\( p^{3} T^{9} + 13296340874080788262 p^{6} T^{10} - 14385104804172608 p^{9} T^{11} + 13842945944569 p^{12} T^{12} - 11058483081 p^{15} T^{13} + 7263714 p^{18} T^{14} - 3319 p^{21} T^{15} + p^{24} T^{16} \)
83 \( 1 - 2057 T + 3428337 T^{2} - 58113074 p T^{3} + 5848930049523 T^{4} - 6127839958694065 T^{5} + 5963805445300349519 T^{6} - \)\(51\!\cdots\!20\)\( T^{7} + \)\(40\!\cdots\!84\)\( T^{8} - \)\(51\!\cdots\!20\)\( p^{3} T^{9} + 5963805445300349519 p^{6} T^{10} - 6127839958694065 p^{9} T^{11} + 5848930049523 p^{12} T^{12} - 58113074 p^{16} T^{13} + 3428337 p^{18} T^{14} - 2057 p^{21} T^{15} + p^{24} T^{16} \)
89 \( 1 - 1668 T + 28635 p T^{2} - 2210533246 T^{3} + 1920473186869 T^{4} - 1573723451807446 T^{5} + 1908156844420284881 T^{6} - \)\(19\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!44\)\( T^{8} - \)\(19\!\cdots\!88\)\( p^{3} T^{9} + 1908156844420284881 p^{6} T^{10} - 1573723451807446 p^{9} T^{11} + 1920473186869 p^{12} T^{12} - 2210533246 p^{15} T^{13} + 28635 p^{19} T^{14} - 1668 p^{21} T^{15} + p^{24} T^{16} \)
97 \( 1 - 1956 T + 3791820 T^{2} - 5487429810 T^{3} + 7196989126723 T^{4} - 7796244231410152 T^{5} + 8593161149310304148 T^{6} - \)\(84\!\cdots\!30\)\( T^{7} + \)\(81\!\cdots\!40\)\( T^{8} - \)\(84\!\cdots\!30\)\( p^{3} T^{9} + 8593161149310304148 p^{6} T^{10} - 7796244231410152 p^{9} T^{11} + 7196989126723 p^{12} T^{12} - 5487429810 p^{15} T^{13} + 3791820 p^{18} T^{14} - 1956 p^{21} T^{15} + p^{24} T^{16} \)
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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

−5.24818082800834043213733079606, −4.91729102188014468784388875909, −4.89028542887256936587192330219, −4.85322532971500373509642428321, −4.69557533441361376736538183325, −4.67173796312431542786644436872, −4.65582734937962653102870568504, −4.34699508255869451716745833988, −4.21746566999097889094382420512, −3.76182584547269052236889615909, −3.65393940995306566276635447984, −3.57837690729454845491835046412, −3.34833005574944957421137828402, −3.30402882192459559539321681417, −3.08308300835040928772366291778, −2.32616077856440584250832541071, −2.20894023297450445440531222765, −1.92412853782076450916311592169, −1.62018724410396158492005331826, −1.26446824158223135992006581756, −1.07232283918971311365301094771, −0.76509155334516902874932818328, −0.70692408628685967928988057097, −0.68288059924350781120312986834, −0.51679212708775744164073075212, 0.51679212708775744164073075212, 0.68288059924350781120312986834, 0.70692408628685967928988057097, 0.76509155334516902874932818328, 1.07232283918971311365301094771, 1.26446824158223135992006581756, 1.62018724410396158492005331826, 1.92412853782076450916311592169, 2.20894023297450445440531222765, 2.32616077856440584250832541071, 3.08308300835040928772366291778, 3.30402882192459559539321681417, 3.34833005574944957421137828402, 3.57837690729454845491835046412, 3.65393940995306566276635447984, 3.76182584547269052236889615909, 4.21746566999097889094382420512, 4.34699508255869451716745833988, 4.65582734937962653102870568504, 4.67173796312431542786644436872, 4.69557533441361376736538183325, 4.85322532971500373509642428321, 4.89028542887256936587192330219, 4.91729102188014468784388875909, 5.24818082800834043213733079606

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

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