Dirichlet series
| L(s) = 1 | + 36·7-s + 19·9-s − 72·11-s + 88·17-s + 144·19-s − 20·23-s + 458·25-s − 144·27-s − 484·29-s − 996·37-s − 156·41-s + 504·43-s + 423·49-s − 1.16e3·53-s − 600·59-s − 1.22e3·61-s + 684·63-s − 960·67-s + 2.96e3·71-s − 2.59e3·77-s − 3.96e3·79-s − 380·81-s − 5.43e3·89-s + 3.04e3·97-s − 1.36e3·99-s + 1.40e3·101-s + 1.40e3·103-s + ⋯ |
| L(s) = 1 | + 1.94·7-s + 0.703·9-s − 1.97·11-s + 1.25·17-s + 1.73·19-s − 0.181·23-s + 3.66·25-s − 1.02·27-s − 3.09·29-s − 4.42·37-s − 0.594·41-s + 1.78·43-s + 1.23·49-s − 3.01·53-s − 1.32·59-s − 2.56·61-s + 1.36·63-s − 1.75·67-s + 4.95·71-s − 3.83·77-s − 5.65·79-s − 0.521·81-s − 6.46·89-s + 3.18·97-s − 1.38·99-s + 1.37·101-s + 1.34·103-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.40474\times 10^{12}\) |
| Root analytic conductor: | \(6.31548\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 13^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.602217339\) |
| \(L(\frac12)\) | \(\approx\) | \(1.602217339\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 19 T^{2} + 16 p^{2} T^{3} + 247 p T^{4} - 368 p^{2} T^{5} + 40106 T^{6} + 11248 p^{2} T^{7} - 790178 T^{8} + 11248 p^{5} T^{9} + 40106 p^{6} T^{10} - 368 p^{11} T^{11} + 247 p^{13} T^{12} + 16 p^{17} T^{13} - 19 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 - 458 T^{2} + 125897 T^{4} - 24160794 T^{6} + 3438237956 T^{8} - 24160794 p^{6} T^{10} + 125897 p^{12} T^{12} - 458 p^{18} T^{14} + p^{24} T^{16} \) | |
| 7 | \( 1 - 36 T + 873 T^{2} - 324 p^{2} T^{3} + 201893 T^{4} - 8952 p^{2} T^{5} - 95093142 T^{6} + 3113286576 T^{7} - 67152988962 T^{8} + 3113286576 p^{3} T^{9} - 95093142 p^{6} T^{10} - 8952 p^{11} T^{11} + 201893 p^{12} T^{12} - 324 p^{17} T^{13} + 873 p^{18} T^{14} - 36 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 72 T + 4541 T^{2} + 202536 T^{3} + 6717133 T^{4} + 240510816 T^{5} + 7948475714 T^{6} + 297590198976 T^{7} + 12294936551566 T^{8} + 297590198976 p^{3} T^{9} + 7948475714 p^{6} T^{10} + 240510816 p^{9} T^{11} + 6717133 p^{12} T^{12} + 202536 p^{15} T^{13} + 4541 p^{18} T^{14} + 72 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 88 T - 9530 T^{2} + 743472 T^{3} + 74278793 T^{4} - 3196501792 T^{5} - 525330591210 T^{6} + 238340891720 p T^{7} + 3262191249902068 T^{8} + 238340891720 p^{4} T^{9} - 525330591210 p^{6} T^{10} - 3196501792 p^{9} T^{11} + 74278793 p^{12} T^{12} + 743472 p^{15} T^{13} - 9530 p^{18} T^{14} - 88 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 144 T + 28129 T^{2} - 3055248 T^{3} + 391400137 T^{4} - 42042636144 T^{5} + 4126488684550 T^{6} - 393418133121024 T^{7} + 31476709399482262 T^{8} - 393418133121024 p^{3} T^{9} + 4126488684550 p^{6} T^{10} - 42042636144 p^{9} T^{11} + 391400137 p^{12} T^{12} - 3055248 p^{15} T^{13} + 28129 p^{18} T^{14} - 144 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 20 T - 13055 T^{2} + 3522516 T^{3} + 136421117 T^{4} - 44765515096 T^{5} + 5987613650034 T^{6} + 482965802453776 T^{7} - 73517605134586610 T^{8} + 482965802453776 p^{3} T^{9} + 5987613650034 p^{6} T^{10} - 44765515096 p^{9} T^{11} + 136421117 p^{12} T^{12} + 3522516 p^{15} T^{13} - 13055 p^{18} T^{14} + 20 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 484 T + 62398 T^{2} + 3102120 T^{3} + 2861699321 T^{4} + 796082031160 T^{5} + 81241366702014 T^{6} + 12932956799673068 T^{7} + 2991182380753721860 T^{8} + 12932956799673068 p^{3} T^{9} + 81241366702014 p^{6} T^{10} + 796082031160 p^{9} T^{11} + 2861699321 p^{12} T^{12} + 3102120 p^{15} T^{13} + 62398 p^{18} T^{14} + 484 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 6840 p T^{2} + 20378911388 T^{4} - 1157065279485432 T^{6} + 42405269588176977990 T^{8} - 1157065279485432 p^{6} T^{10} + 20378911388 p^{12} T^{12} - 6840 p^{19} T^{14} + p^{24} T^{16} \) | |
| 37 | \( 1 + 996 T + 539310 T^{2} + 207803448 T^{3} + 59330251721 T^{4} + 12346099294680 T^{5} + 1767383612627598 T^{6} + 130153579648604508 T^{7} + 1804763830510508100 T^{8} + 130153579648604508 p^{3} T^{9} + 1767383612627598 p^{6} T^{10} + 12346099294680 p^{9} T^{11} + 59330251721 p^{12} T^{12} + 207803448 p^{15} T^{13} + 539310 p^{18} T^{14} + 996 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 + 156 T + 152230 T^{2} + 22482408 T^{3} + 11298512945 T^{4} - 1853522430936 T^{5} + 101999819664822 T^{6} - 359733416454103308 T^{7} - 45504645499627025212 T^{8} - 359733416454103308 p^{3} T^{9} + 101999819664822 p^{6} T^{10} - 1853522430936 p^{9} T^{11} + 11298512945 p^{12} T^{12} + 22482408 p^{15} T^{13} + 152230 p^{18} T^{14} + 156 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 504 T - 62831 T^{2} + 34404072 T^{3} + 15300163321 T^{4} - 1995789294576 T^{5} - 1928319302072330 T^{6} + 115612138867794816 T^{7} + \)\(14\!\cdots\!58\)\( T^{8} + 115612138867794816 p^{3} T^{9} - 1928319302072330 p^{6} T^{10} - 1995789294576 p^{9} T^{11} + 15300163321 p^{12} T^{12} + 34404072 p^{15} T^{13} - 62831 p^{18} T^{14} - 504 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 - 591496 T^{2} + 173108714140 T^{4} - 31704149230284472 T^{6} + \)\(39\!\cdots\!70\)\( T^{8} - 31704149230284472 p^{6} T^{10} + 173108714140 p^{12} T^{12} - 591496 p^{18} T^{14} + p^{24} T^{16} \) | |
| 53 | \( ( 1 + 582 T + 506693 T^{2} + 252459774 T^{3} + 107034081780 T^{4} + 252459774 p^{3} T^{5} + 506693 p^{6} T^{6} + 582 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 59 | \( 1 + 600 T + 581701 T^{2} + 277020600 T^{3} + 136123467869 T^{4} + 70329728787648 T^{5} + 35782673050162002 T^{6} + 19313701908153367488 T^{7} + \)\(97\!\cdots\!10\)\( T^{8} + 19313701908153367488 p^{3} T^{9} + 35782673050162002 p^{6} T^{10} + 70329728787648 p^{9} T^{11} + 136123467869 p^{12} T^{12} + 277020600 p^{15} T^{13} + 581701 p^{18} T^{14} + 600 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 1224 T + 500350 T^{2} + 35614512 T^{3} - 35822730671 T^{4} - 22495862931552 T^{5} - 15560623274893634 T^{6} - 10007556558962344056 T^{7} - \)\(51\!\cdots\!32\)\( T^{8} - 10007556558962344056 p^{3} T^{9} - 15560623274893634 p^{6} T^{10} - 22495862931552 p^{9} T^{11} - 35822730671 p^{12} T^{12} + 35614512 p^{15} T^{13} + 500350 p^{18} T^{14} + 1224 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 960 T + 1280625 T^{2} + 934488000 T^{3} + 803908892825 T^{4} + 562420802431776 T^{5} + 376723450854000438 T^{6} + \)\(23\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!98\)\( T^{8} + \)\(23\!\cdots\!60\)\( p^{3} T^{9} + 376723450854000438 p^{6} T^{10} + 562420802431776 p^{9} T^{11} + 803908892825 p^{12} T^{12} + 934488000 p^{15} T^{13} + 1280625 p^{18} T^{14} + 960 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 2964 T + 4460069 T^{2} - 4539772068 T^{3} + 3351908178577 T^{4} - 1759904581589784 T^{5} + 554975332339663694 T^{6} + 4576872542816618256 T^{7} - \)\(91\!\cdots\!46\)\( T^{8} + 4576872542816618256 p^{3} T^{9} + 554975332339663694 p^{6} T^{10} - 1759904581589784 p^{9} T^{11} + 3351908178577 p^{12} T^{12} - 4539772068 p^{15} T^{13} + 4460069 p^{18} T^{14} - 2964 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 1897314 T^{2} + 1657159865729 T^{4} - 922036132562901858 T^{6} + \)\(39\!\cdots\!36\)\( T^{8} - 922036132562901858 p^{6} T^{10} + 1657159865729 p^{12} T^{12} - 1897314 p^{18} T^{14} + p^{24} T^{16} \) | |
| 79 | \( ( 1 + 1984 T + 2636300 T^{2} + 2535484224 T^{3} + 2000706118694 T^{4} + 2535484224 p^{3} T^{5} + 2636300 p^{6} T^{6} + 1984 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 83 | \( 1 - 3370304 T^{2} + 5186626158236 T^{4} - 4932043412107063488 T^{6} + \)\(32\!\cdots\!90\)\( T^{8} - 4932043412107063488 p^{6} T^{10} + 5186626158236 p^{12} T^{12} - 3370304 p^{18} T^{14} + p^{24} T^{16} \) | |
| 89 | \( 1 + 5430 T + 15216971 T^{2} + 29260483530 T^{3} + 43216265090809 T^{4} + 52483952790895932 T^{5} + 55092174085554978782 T^{6} + \)\(51\!\cdots\!72\)\( T^{7} + \)\(45\!\cdots\!30\)\( T^{8} + \)\(51\!\cdots\!72\)\( p^{3} T^{9} + 55092174085554978782 p^{6} T^{10} + 52483952790895932 p^{9} T^{11} + 43216265090809 p^{12} T^{12} + 29260483530 p^{15} T^{13} + 15216971 p^{18} T^{14} + 5430 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 3042 T + 6393187 T^{2} - 10064758158 T^{3} + 12686093981641 T^{4} - 13372299704167092 T^{5} + 12676544447228147374 T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!66\)\( T^{8} - \)\(11\!\cdots\!20\)\( p^{3} T^{9} + 12676544447228147374 p^{6} T^{10} - 13372299704167092 p^{9} T^{11} + 12686093981641 p^{12} T^{12} - 10064758158 p^{15} T^{13} + 6393187 p^{18} T^{14} - 3042 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.14955588386999190836294225998, −3.87518529971314227295923782434, −3.84587704250978049492441978966, −3.67333094101116616793529336574, −3.59122986071165173302301499814, −3.35519904856663360179620380746, −3.24943062754216605779366214988, −2.95339217347809490428583889228, −2.93688347900668464334945975687, −2.91687611855608649260119950922, −2.75792967628977186145427521378, −2.61301010379728335144655207653, −2.35822722212395946015183121260, −2.13982884464082795710906050424, −1.77045518976126340279817302457, −1.71189687765858880712708046581, −1.64439162536418291099018094310, −1.43212122283271345947890710183, −1.38629942796279833672291346934, −1.34476186519058651489196418988, −1.22888621678318465323230646385, −0.55998222487908855548811100885, −0.43573963739093531227121001725, −0.43567162025901435878848488145, −0.084007303423178931087757874458, 0.084007303423178931087757874458, 0.43567162025901435878848488145, 0.43573963739093531227121001725, 0.55998222487908855548811100885, 1.22888621678318465323230646385, 1.34476186519058651489196418988, 1.38629942796279833672291346934, 1.43212122283271345947890710183, 1.64439162536418291099018094310, 1.71189687765858880712708046581, 1.77045518976126340279817302457, 2.13982884464082795710906050424, 2.35822722212395946015183121260, 2.61301010379728335144655207653, 2.75792967628977186145427521378, 2.91687611855608649260119950922, 2.93688347900668464334945975687, 2.95339217347809490428583889228, 3.24943062754216605779366214988, 3.35519904856663360179620380746, 3.59122986071165173302301499814, 3.67333094101116616793529336574, 3.84587704250978049492441978966, 3.87518529971314227295923782434, 4.14955588386999190836294225998
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.