Dirichlet series
| L(s) = 1 | + 4·2-s + 2·3-s − 40·5-s + 8·6-s − 32·7-s − 44·8-s − 25·9-s − 160·10-s + 44·11-s + 86·13-s − 128·14-s − 80·15-s − 121·16-s − 100·18-s + 288·19-s − 64·21-s + 176·22-s − 160·23-s − 88·24-s + 900·25-s + 344·26-s − 34·27-s + 642·29-s − 320·30-s + 182·31-s − 16·32-s + 88·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s − 3.57·5-s + 0.544·6-s − 1.72·7-s − 1.94·8-s − 0.925·9-s − 5.05·10-s + 1.20·11-s + 1.83·13-s − 2.44·14-s − 1.37·15-s − 1.89·16-s − 1.30·18-s + 3.47·19-s − 0.665·21-s + 1.70·22-s − 1.45·23-s − 0.748·24-s + 36/5·25-s + 2.59·26-s − 0.242·27-s + 4.11·29-s − 1.94·30-s + 1.05·31-s − 0.0883·32-s + 0.464·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.79171\times 10^{15}\) |
| Root analytic conductor: | \(9.23351\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 17^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(3.959262021\) |
| \(L(\frac12)\) | \(\approx\) | \(3.959262021\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p T )^{8} \) |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T + p^{4} T^{2} - 5 p^{2} T^{3} + 25 T^{4} + 17 p^{3} T^{5} - 85 p T^{6} - 17 p^{4} T^{7} + 61 p^{6} T^{8} - 17 p^{7} T^{9} - 85 p^{7} T^{10} + 17 p^{12} T^{11} + 25 p^{12} T^{12} - 5 p^{17} T^{13} + p^{22} T^{14} - p^{23} T^{15} + p^{24} T^{16} \) |
| 3 | \( 1 - 2 T + 29 T^{2} - 74 T^{3} + 1117 T^{4} + 724 T^{5} + 11170 p T^{6} - 3040 p T^{7} + 715922 T^{8} - 3040 p^{4} T^{9} + 11170 p^{7} T^{10} + 724 p^{9} T^{11} + 1117 p^{12} T^{12} - 74 p^{15} T^{13} + 29 p^{18} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 + 32 T + 1283 T^{2} + 32252 T^{3} + 1012561 T^{4} + 21378740 T^{5} + 530593074 T^{6} + 1407417924 p T^{7} + 210970433494 T^{8} + 1407417924 p^{4} T^{9} + 530593074 p^{6} T^{10} + 21378740 p^{9} T^{11} + 1012561 p^{12} T^{12} + 32252 p^{15} T^{13} + 1283 p^{18} T^{14} + 32 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 - 4 p T + 283 p T^{2} - 81662 T^{3} + 7583982 T^{4} - 247447294 T^{5} + 15298441611 T^{6} - 33863216716 p T^{7} + 20706399769338 T^{8} - 33863216716 p^{4} T^{9} + 15298441611 p^{6} T^{10} - 247447294 p^{9} T^{11} + 7583982 p^{12} T^{12} - 81662 p^{15} T^{13} + 283 p^{19} T^{14} - 4 p^{22} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 86 T + 10172 T^{2} - 370166 T^{3} + 27392824 T^{4} - 418396946 T^{5} + 72505272436 T^{6} - 2024661411362 T^{7} + 234372774380430 T^{8} - 2024661411362 p^{3} T^{9} + 72505272436 p^{6} T^{10} - 418396946 p^{9} T^{11} + 27392824 p^{12} T^{12} - 370166 p^{15} T^{13} + 10172 p^{18} T^{14} - 86 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 288 T + 60385 T^{2} - 24458 p^{2} T^{3} + 1095228838 T^{4} - 118999107066 T^{5} + 11895084348691 T^{6} - 1119777595123400 T^{7} + 5052560930680494 p T^{8} - 1119777595123400 p^{3} T^{9} + 11895084348691 p^{6} T^{10} - 118999107066 p^{9} T^{11} + 1095228838 p^{12} T^{12} - 24458 p^{17} T^{13} + 60385 p^{18} T^{14} - 288 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 160 T + 56451 T^{2} + 6853788 T^{3} + 1495940913 T^{4} + 147401498276 T^{5} + 26173587326354 T^{6} + 2223400672175900 T^{7} + 351938603447093622 T^{8} + 2223400672175900 p^{3} T^{9} + 26173587326354 p^{6} T^{10} + 147401498276 p^{9} T^{11} + 1495940913 p^{12} T^{12} + 6853788 p^{15} T^{13} + 56451 p^{18} T^{14} + 160 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 642 T + 313893 T^{2} - 106994538 T^{3} + 31322085262 T^{4} - 7541550943934 T^{5} + 1619604804567043 T^{6} - 299781406251811254 T^{7} + 50139298523971690146 T^{8} - 299781406251811254 p^{3} T^{9} + 1619604804567043 p^{6} T^{10} - 7541550943934 p^{9} T^{11} + 31322085262 p^{12} T^{12} - 106994538 p^{15} T^{13} + 313893 p^{18} T^{14} - 642 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 182 T + 85576 T^{2} - 10716770 T^{3} + 4140804024 T^{4} - 591660081986 T^{5} + 162802029428520 T^{6} - 20651357537993302 T^{7} + 4752825669667304622 T^{8} - 20651357537993302 p^{3} T^{9} + 162802029428520 p^{6} T^{10} - 591660081986 p^{9} T^{11} + 4140804024 p^{12} T^{12} - 10716770 p^{15} T^{13} + 85576 p^{18} T^{14} - 182 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 490 T + 201728 T^{2} + 65100122 T^{3} + 24042875328 T^{4} + 6643040322414 T^{5} + 1818874834053248 T^{6} + 440140940629597326 T^{7} + \)\(11\!\cdots\!34\)\( T^{8} + 440140940629597326 p^{3} T^{9} + 1818874834053248 p^{6} T^{10} + 6643040322414 p^{9} T^{11} + 24042875328 p^{12} T^{12} + 65100122 p^{15} T^{13} + 201728 p^{18} T^{14} + 490 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 72 T + 307912 T^{2} - 41048632 T^{3} + 1030792382 p T^{4} - 9270482206352 T^{5} + 3607469054834904 T^{6} - 1104681669900989408 T^{7} + \)\(25\!\cdots\!35\)\( T^{8} - 1104681669900989408 p^{3} T^{9} + 3607469054834904 p^{6} T^{10} - 9270482206352 p^{9} T^{11} + 1030792382 p^{13} T^{12} - 41048632 p^{15} T^{13} + 307912 p^{18} T^{14} - 72 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 1168 T + 1008368 T^{2} - 610733040 T^{3} + 315537402348 T^{4} - 135072162470704 T^{5} + 51637117850596176 T^{6} - 17153282405281325008 T^{7} + \)\(12\!\cdots\!14\)\( p T^{8} - 17153282405281325008 p^{3} T^{9} + 51637117850596176 p^{6} T^{10} - 135072162470704 p^{9} T^{11} + 315537402348 p^{12} T^{12} - 610733040 p^{15} T^{13} + 1008368 p^{18} T^{14} - 1168 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 - 216 T + 318508 T^{2} - 83458024 T^{3} + 70373832484 T^{4} - 15196407035304 T^{5} + 10781484044393044 T^{6} - 2160366376548699544 T^{7} + \)\(12\!\cdots\!98\)\( T^{8} - 2160366376548699544 p^{3} T^{9} + 10781484044393044 p^{6} T^{10} - 15196407035304 p^{9} T^{11} + 70373832484 p^{12} T^{12} - 83458024 p^{15} T^{13} + 318508 p^{18} T^{14} - 216 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 508 T + 534536 T^{2} - 320092212 T^{3} + 157216139644 T^{4} - 87718299300908 T^{5} + 38067522178603768 T^{6} - 15358719451707244740 T^{7} + \)\(69\!\cdots\!42\)\( T^{8} - 15358719451707244740 p^{3} T^{9} + 38067522178603768 p^{6} T^{10} - 87718299300908 p^{9} T^{11} + 157216139644 p^{12} T^{12} - 320092212 p^{15} T^{13} + 534536 p^{18} T^{14} - 508 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 918 T + 1282589 T^{2} + 706235826 T^{3} + 534690695978 T^{4} + 167571319625306 T^{5} + 93037330694967019 T^{6} + 10157650820458665950 T^{7} + \)\(11\!\cdots\!30\)\( T^{8} + 10157650820458665950 p^{3} T^{9} + 93037330694967019 p^{6} T^{10} + 167571319625306 p^{9} T^{11} + 534690695978 p^{12} T^{12} + 706235826 p^{15} T^{13} + 1282589 p^{18} T^{14} + 918 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 1970 T + 2883565 T^{2} - 49790794 p T^{3} + 2705911309662 T^{4} - 2009854585936118 T^{5} + 1314517958944254795 T^{6} - \)\(74\!\cdots\!78\)\( T^{7} + \)\(38\!\cdots\!70\)\( T^{8} - \)\(74\!\cdots\!78\)\( p^{3} T^{9} + 1314517958944254795 p^{6} T^{10} - 2009854585936118 p^{9} T^{11} + 2705911309662 p^{12} T^{12} - 49790794 p^{16} T^{13} + 2883565 p^{18} T^{14} - 1970 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 2750 T + 5181377 T^{2} - 6916261226 T^{3} + 7561002543697 T^{4} - 6803559310615928 T^{5} + 5286006372279497374 T^{6} - \)\(35\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!50\)\( T^{8} - \)\(35\!\cdots\!80\)\( p^{3} T^{9} + 5286006372279497374 p^{6} T^{10} - 6803559310615928 p^{9} T^{11} + 7561002543697 p^{12} T^{12} - 6916261226 p^{15} T^{13} + 5181377 p^{18} T^{14} - 2750 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1368 T + 32947 p T^{2} - 2191829806 T^{3} + 2250521403258 T^{4} - 1662710827205582 T^{5} + 1317349584645401615 T^{6} - \)\(82\!\cdots\!40\)\( T^{7} + \)\(54\!\cdots\!74\)\( T^{8} - \)\(82\!\cdots\!40\)\( p^{3} T^{9} + 1317349584645401615 p^{6} T^{10} - 1662710827205582 p^{9} T^{11} + 2250521403258 p^{12} T^{12} - 2191829806 p^{15} T^{13} + 32947 p^{19} T^{14} - 1368 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 582 T + 1831676 T^{2} - 1248086638 T^{3} + 1743479561400 T^{4} - 1217822600170690 T^{5} + 1116177973236591412 T^{6} - \)\(71\!\cdots\!98\)\( T^{7} + \)\(51\!\cdots\!46\)\( T^{8} - \)\(71\!\cdots\!98\)\( p^{3} T^{9} + 1116177973236591412 p^{6} T^{10} - 1217822600170690 p^{9} T^{11} + 1743479561400 p^{12} T^{12} - 1248086638 p^{15} T^{13} + 1831676 p^{18} T^{14} - 582 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 1256 T + 2294421 T^{2} - 1229570338 T^{3} + 1353095962778 T^{4} - 20521948234058 T^{5} + 337078523511320983 T^{6} + \)\(28\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} + \)\(28\!\cdots\!16\)\( p^{3} T^{9} + 337078523511320983 p^{6} T^{10} - 20521948234058 p^{9} T^{11} + 1353095962778 p^{12} T^{12} - 1229570338 p^{15} T^{13} + 2294421 p^{18} T^{14} - 1256 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 2534 T + 4410181 T^{2} - 5702579186 T^{3} + 6026070763785 T^{4} - 5526722493075392 T^{5} + 4559876382247100950 T^{6} - \)\(35\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!70\)\( T^{8} - \)\(35\!\cdots\!20\)\( p^{3} T^{9} + 4559876382247100950 p^{6} T^{10} - 5526722493075392 p^{9} T^{11} + 6026070763785 p^{12} T^{12} - 5702579186 p^{15} T^{13} + 4410181 p^{18} T^{14} - 2534 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 1200 T + 3314380 T^{2} - 3458738320 T^{3} + 5922068541106 T^{4} - 5417018940322848 T^{5} + 6875372653271459104 T^{6} - \)\(54\!\cdots\!28\)\( T^{7} + \)\(57\!\cdots\!47\)\( T^{8} - \)\(54\!\cdots\!28\)\( p^{3} T^{9} + 6875372653271459104 p^{6} T^{10} - 5417018940322848 p^{9} T^{11} + 5922068541106 p^{12} T^{12} - 3458738320 p^{15} T^{13} + 3314380 p^{18} T^{14} - 1200 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 1698 T + 6364400 T^{2} + 8712212370 T^{3} + 18263894634424 T^{4} + 20455242697131230 T^{5} + 31089619215697014208 T^{6} + \)\(28\!\cdots\!30\)\( T^{7} + \)\(34\!\cdots\!38\)\( T^{8} + \)\(28\!\cdots\!30\)\( p^{3} T^{9} + 31089619215697014208 p^{6} T^{10} + 20455242697131230 p^{9} T^{11} + 18263894634424 p^{12} T^{12} + 8712212370 p^{15} T^{13} + 6364400 p^{18} T^{14} + 1698 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−3.72405116117659797387169931018, −3.56114599859228502509499416037, −3.42246930017743483976029319394, −3.38719825522719364651300482243, −3.36514877090492704981077074053, −3.05301811769733971145541101856, −2.96876699084541972083572979410, −2.72504942671428090292500234860, −2.72057477662686844688866860661, −2.67855502921745114393149845751, −2.50514737139086232076545907483, −2.48359745429594197218572258500, −1.99397435264505536605033633970, −1.98032045730439164040850361900, −1.93346486497071022200391386310, −1.26008885484155242848379963901, −1.25226681015453440170635485980, −1.24662039667085840375068708432, −0.815012797217917887710991974023, −0.801025611064115405256060537452, −0.76501060244635937137648952807, −0.71478710862391281392109981225, −0.51358894669946963121213817615, −0.43052693711502763806522014396, −0.081398477882163589680451254129, 0.081398477882163589680451254129, 0.43052693711502763806522014396, 0.51358894669946963121213817615, 0.71478710862391281392109981225, 0.76501060244635937137648952807, 0.801025611064115405256060537452, 0.815012797217917887710991974023, 1.24662039667085840375068708432, 1.25226681015453440170635485980, 1.26008885484155242848379963901, 1.93346486497071022200391386310, 1.98032045730439164040850361900, 1.99397435264505536605033633970, 2.48359745429594197218572258500, 2.50514737139086232076545907483, 2.67855502921745114393149845751, 2.72057477662686844688866860661, 2.72504942671428090292500234860, 2.96876699084541972083572979410, 3.05301811769733971145541101856, 3.36514877090492704981077074053, 3.38719825522719364651300482243, 3.42246930017743483976029319394, 3.56114599859228502509499416037, 3.72405116117659797387169931018
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.