Dirichlet series
| L(s) = 1 | + 36·7-s + 19·9-s − 72·11-s + 62·13-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 + 2.23e3·91-s − 3.04e3·97-s − 1.36e3·99-s + ⋯ |
| L(s) = 1 | + 1.94·7-s + 0.703·9-s − 1.97·11-s + 1.32·13-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 + 2.57·91-s − 3.18·97-s − 1.38·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.14559\times 10^{8}\) |
| Root analytic conductor: | \(3.50319\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 13^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(20.70070450\) |
| \(L(\frac12)\) | \(\approx\) | \(20.70070450\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 - 62 T - 115 T^{2} + 7554 p T^{3} - 31708 p^{2} T^{4} + 7554 p^{4} T^{5} - 115 p^{6} T^{6} - 62 p^{9} T^{7} + p^{12} T^{8} \) | |
| 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
−4.97725338034322706761749980549, −4.92828991440122013700299973752, −4.88682441038509132995073802393, −4.74131485290694425388894568392, −4.57752261818654201396287562946, −4.52534230869626352594552367796, −4.22140234371288922042313135056, −3.86556646300532651671054522956, −3.77303539958083690590864569837, −3.37151123932651122122777352195, −3.31714358629335215137300058145, −3.28662367707533891565821997245, −3.23297473515889214180028689634, −3.03917293735205067185704385152, −2.48039521217116168990388809661, −2.41160676820982834046638963000, −2.19960484714657884427254322398, −2.17588029323835282059604188132, −1.75116894806963877906961369489, −1.31772445263290893038022461808, −1.23351210668319516596639107811, −1.16463533527834403520848195022, −0.944444297645829075868440817784, −0.52407146000275570176585492377, −0.40509918730244166318724374059, 0.40509918730244166318724374059, 0.52407146000275570176585492377, 0.944444297645829075868440817784, 1.16463533527834403520848195022, 1.23351210668319516596639107811, 1.31772445263290893038022461808, 1.75116894806963877906961369489, 2.17588029323835282059604188132, 2.19960484714657884427254322398, 2.41160676820982834046638963000, 2.48039521217116168990388809661, 3.03917293735205067185704385152, 3.23297473515889214180028689634, 3.28662367707533891565821997245, 3.31714358629335215137300058145, 3.37151123932651122122777352195, 3.77303539958083690590864569837, 3.86556646300532651671054522956, 4.22140234371288922042313135056, 4.52534230869626352594552367796, 4.57752261818654201396287562946, 4.74131485290694425388894568392, 4.88682441038509132995073802393, 4.92828991440122013700299973752, 4.97725338034322706761749980549
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.