Properties

Label 16-242e8-1.1-c3e8-0-3
Degree $16$
Conductor $1.176\times 10^{19}$
Sign $1$
Analytic cond. $1.72763\times 10^{9}$
Root an. cond. $3.77868$
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  − 4·2-s + 3·3-s + 4·4-s + 5·5-s − 12·6-s + 7-s + 21·9-s − 20·10-s + 12·12-s − 7·13-s − 4·14-s + 15·15-s − 161·17-s − 84·18-s + 272·19-s + 20·20-s + 3·21-s + 628·23-s + 129·25-s + 28·26-s − 59·27-s + 4·28-s − 33·29-s − 60·30-s + 323·31-s + 64·32-s + 644·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.0539·7-s + 7/9·9-s − 0.632·10-s + 0.288·12-s − 0.149·13-s − 0.0763·14-s + 0.258·15-s − 2.29·17-s − 1.09·18-s + 3.28·19-s + 0.223·20-s + 0.0311·21-s + 5.69·23-s + 1.03·25-s + 0.211·26-s − 0.420·27-s + 0.0269·28-s − 0.211·29-s − 0.365·30-s + 1.87·31-s + 0.353·32-s + 3.24·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(1.72763\times 10^{9}\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 11^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3013988045\)
\(L(\frac12)\) \(\approx\) \(0.3013988045\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
11 \( 1 \)
good3 \( 1 - p T - 4 p T^{2} + 158 T^{3} - 1228 T^{4} + 6473 T^{5} + 959 T^{6} - 225344 T^{7} + 1236232 T^{8} - 225344 p^{3} T^{9} + 959 p^{6} T^{10} + 6473 p^{9} T^{11} - 1228 p^{12} T^{12} + 158 p^{15} T^{13} - 4 p^{19} T^{14} - p^{22} T^{15} + p^{24} T^{16} \)
5 \( 1 - p T - 104 T^{2} - 326 p T^{3} + 10006 T^{4} + 56777 p T^{5} + 2835571 T^{6} - 1265534 p^{2} T^{7} - 387953844 T^{8} - 1265534 p^{5} T^{9} + 2835571 p^{6} T^{10} + 56777 p^{10} T^{11} + 10006 p^{12} T^{12} - 326 p^{16} T^{13} - 104 p^{18} T^{14} - p^{22} T^{15} + p^{24} T^{16} \)
7 \( 1 - T + 12 T^{2} + 1104 T^{3} - 87888 T^{4} + 922401 T^{5} + 28062801 T^{6} - 211635072 T^{7} + 13215739812 T^{8} - 211635072 p^{3} T^{9} + 28062801 p^{6} T^{10} + 922401 p^{9} T^{11} - 87888 p^{12} T^{12} + 1104 p^{15} T^{13} + 12 p^{18} T^{14} - p^{21} T^{15} + p^{24} T^{16} \)
13 \( 1 + 7 T + 1930 T^{2} - 79384 T^{3} + 10224596 T^{4} - 326819897 T^{5} + 769971197 p T^{6} - 1235524128730 T^{7} + 47709276128944 T^{8} - 1235524128730 p^{3} T^{9} + 769971197 p^{7} T^{10} - 326819897 p^{9} T^{11} + 10224596 p^{12} T^{12} - 79384 p^{15} T^{13} + 1930 p^{18} T^{14} + 7 p^{21} T^{15} + p^{24} T^{16} \)
17 \( 1 + 161 T + 5020 T^{2} - 211188 T^{3} + 21130156 T^{4} + 1323457999 T^{5} - 265862283281 T^{6} - 778694329080 p T^{7} + 277440196391544 T^{8} - 778694329080 p^{4} T^{9} - 265862283281 p^{6} T^{10} + 1323457999 p^{9} T^{11} + 21130156 p^{12} T^{12} - 211188 p^{15} T^{13} + 5020 p^{18} T^{14} + 161 p^{21} T^{15} + p^{24} T^{16} \)
19 \( 1 - 272 T + 33201 T^{2} - 3039584 T^{3} + 292957015 T^{4} - 21884364992 T^{5} + 909384182595 T^{6} - 32108296725040 T^{7} + 2715838419478696 T^{8} - 32108296725040 p^{3} T^{9} + 909384182595 p^{6} T^{10} - 21884364992 p^{9} T^{11} + 292957015 p^{12} T^{12} - 3039584 p^{15} T^{13} + 33201 p^{18} T^{14} - 272 p^{21} T^{15} + p^{24} T^{16} \)
23 \( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} - 314 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
29 \( 1 + 33 T + 14176 T^{2} - 2091144 T^{3} + 419909220 T^{4} - 81926257017 T^{5} + 7198320043835 T^{6} - 2719607629962300 T^{7} + 63828590693142176 T^{8} - 2719607629962300 p^{3} T^{9} + 7198320043835 p^{6} T^{10} - 81926257017 p^{9} T^{11} + 419909220 p^{12} T^{12} - 2091144 p^{15} T^{13} + 14176 p^{18} T^{14} + 33 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 1165032156515 p T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - 3925905573337100 p^{3} T^{9} + 1165032156515 p^{7} T^{10} - 93424183073 p^{9} T^{11} + 305184598 p^{12} T^{12} + 5799192 p^{15} T^{13} - 4098 p^{18} T^{14} - 323 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - 14416097444880138 p^{3} T^{9} + 91349107260671 p^{6} T^{10} + 1042572020189 p^{9} T^{11} - 566286138 p^{12} T^{12} + 4260746 p^{15} T^{13} - 7868 p^{18} T^{14} - 49 p^{21} T^{15} + p^{24} T^{16} \)
41 \( 1 + 361 T - 19140 T^{2} - 12862580 T^{3} + 3853225260 T^{4} + 605378813703 T^{5} - 283363657590617 T^{6} - 99783171614693080 T^{7} - 22290482644766813160 T^{8} - 99783171614693080 p^{3} T^{9} - 283363657590617 p^{6} T^{10} + 605378813703 p^{9} T^{11} + 3853225260 p^{12} T^{12} - 12862580 p^{15} T^{13} - 19140 p^{18} T^{14} + 361 p^{21} T^{15} + p^{24} T^{16} \)
43 \( ( 1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} + 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( 1 + 1069 T + 301070 T^{2} - 43368042 T^{3} - 24642745634 T^{4} + 6223420821981 T^{5} + 4635059307315839 T^{6} + 746453682627128500 T^{7} + 29698413240018564184 T^{8} + 746453682627128500 p^{3} T^{9} + 4635059307315839 p^{6} T^{10} + 6223420821981 p^{9} T^{11} - 24642745634 p^{12} T^{12} - 43368042 p^{15} T^{13} + 301070 p^{18} T^{14} + 1069 p^{21} T^{15} + p^{24} T^{16} \)
53 \( 1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} + 107096647673733916 p^{3} T^{9} + 8874412003121445 p^{6} T^{10} + 4932564267181 p^{9} T^{11} + 33213391926 p^{12} T^{12} - 60864958 p^{15} T^{13} - 343802 p^{18} T^{14} + 281 p^{21} T^{15} + p^{24} T^{16} \)
59 \( 1 + 128 T - 87119 T^{2} - 50674984 T^{3} + 39669673335 T^{4} + 496570131448 T^{5} - 4049042036547885 T^{6} - 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} - 22790010341236160 p^{3} T^{9} - 4049042036547885 p^{6} T^{10} + 496570131448 p^{9} T^{11} + 39669673335 p^{12} T^{12} - 50674984 p^{15} T^{13} - 87119 p^{18} T^{14} + 128 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 - 617 T - 60798 T^{2} + 112995618 T^{3} + 22253617738 T^{4} - 1960695737 p^{2} T^{5} - 13214872371636415 T^{6} - 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} - 3429112403997460700 p^{3} T^{9} - 13214872371636415 p^{6} T^{10} - 1960695737 p^{11} T^{11} + 22253617738 p^{12} T^{12} + 112995618 p^{15} T^{13} - 60798 p^{18} T^{14} - 617 p^{21} T^{15} + p^{24} T^{16} \)
67 \( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( 1 - 115 T - 138428 T^{2} + 42960000 T^{3} + 77925723838 T^{4} - 71653925274035 T^{5} + 12191505393571699 T^{6} + 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} + 3285417296443548450 p^{3} T^{9} + 12191505393571699 p^{6} T^{10} - 71653925274035 p^{9} T^{11} + 77925723838 p^{12} T^{12} + 42960000 p^{15} T^{13} - 138428 p^{18} T^{14} - 115 p^{21} T^{15} + p^{24} T^{16} \)
73 \( 1 - 1487 T + 142728 T^{2} + 14316264 p T^{3} - 569519703768 T^{4} - 366060407928453 T^{5} + 393291607153115139 T^{6} + 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} + 68019158899540158384 p^{3} T^{9} + 393291607153115139 p^{6} T^{10} - 366060407928453 p^{9} T^{11} - 569519703768 p^{12} T^{12} + 14316264 p^{16} T^{13} + 142728 p^{18} T^{14} - 1487 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 + 71 T - 500222 T^{2} + 73013624 T^{3} + 350790961998 T^{4} - 97550067497959 T^{5} - 161731343227399015 T^{6} - 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} - 8534351363848547400 p^{3} T^{9} - 161731343227399015 p^{6} T^{10} - 97550067497959 p^{9} T^{11} + 350790961998 p^{12} T^{12} + 73013624 p^{15} T^{13} - 500222 p^{18} T^{14} + 71 p^{21} T^{15} + p^{24} T^{16} \)
83 \( 1 + 1942 T + 875395 T^{2} - 857035884 T^{3} - 790010610449 T^{4} + 618864428088728 T^{5} + 845503237658086801 T^{6} - \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} - \)\(17\!\cdots\!30\)\( p^{3} T^{9} + 845503237658086801 p^{6} T^{10} + 618864428088728 p^{9} T^{11} - 790010610449 p^{12} T^{12} - 857035884 p^{15} T^{13} + 875395 p^{18} T^{14} + 1942 p^{21} T^{15} + p^{24} T^{16} \)
89 \( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
97 \( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(96\!\cdots\!12\)\( p^{3} T^{9} - 1475957592224572125 p^{6} T^{10} - 4959875190278012 p^{9} T^{11} + 1479496697751 p^{12} T^{12} + 9500759764 p^{15} T^{13} + 10343177 p^{18} T^{14} + 5128 p^{21} T^{15} + p^{24} T^{16} \)
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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

−5.01317621407192642087967359849, −4.99246268852512271250632908442, −4.76893174015539985555313390781, −4.62910996432490173678394117137, −4.20036064242253993235117429969, −4.10616967473955911277847267268, −3.98381403736901414586373880157, −3.97940451019460234412269786095, −3.47890319263544293980466858536, −3.34680690307376841989866473271, −3.11637567383177645001055206971, −2.95441782250111880581647936573, −2.93351921992928910774226343935, −2.82124773083245876073420481582, −2.68405094146065942677791900436, −2.59197730674599220581679386785, −1.90598950028694218039775997549, −1.57399154039191841147324058917, −1.57279878293086002983449829293, −1.49827896261399512337842589304, −1.44413383215151670598164357319, −0.926534098859471454309355588596, −0.70558055547659924263823878325, −0.53664915294456815128777558007, −0.06317516187483957773038501636, 0.06317516187483957773038501636, 0.53664915294456815128777558007, 0.70558055547659924263823878325, 0.926534098859471454309355588596, 1.44413383215151670598164357319, 1.49827896261399512337842589304, 1.57279878293086002983449829293, 1.57399154039191841147324058917, 1.90598950028694218039775997549, 2.59197730674599220581679386785, 2.68405094146065942677791900436, 2.82124773083245876073420481582, 2.93351921992928910774226343935, 2.95441782250111880581647936573, 3.11637567383177645001055206971, 3.34680690307376841989866473271, 3.47890319263544293980466858536, 3.97940451019460234412269786095, 3.98381403736901414586373880157, 4.10616967473955911277847267268, 4.20036064242253993235117429969, 4.62910996432490173678394117137, 4.76893174015539985555313390781, 4.99246268852512271250632908442, 5.01317621407192642087967359849

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

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