Dirichlet series
| L(s) = 1 | + 8·5-s + 8·11-s − 4·13-s − 32·17-s + 160·19-s + 200·23-s + 278·25-s + 216·29-s − 80·31-s − 552·37-s + 384·41-s − 160·43-s + 768·47-s + 820·49-s − 1.88e3·53-s + 64·55-s + 992·59-s + 548·61-s − 32·65-s − 464·67-s − 3.44e3·71-s − 1.52e3·73-s − 688·79-s + 2.12e3·83-s − 256·85-s − 4.22e3·89-s + 1.28e3·95-s + ⋯ |
| L(s) = 1 | + 0.715·5-s + 0.219·11-s − 0.0853·13-s − 0.456·17-s + 1.93·19-s + 1.81·23-s + 2.22·25-s + 1.38·29-s − 0.463·31-s − 2.45·37-s + 1.46·41-s − 0.567·43-s + 2.38·47-s + 2.39·49-s − 4.89·53-s + 0.156·55-s + 2.18·59-s + 1.15·61-s − 0.0610·65-s − 0.846·67-s − 5.75·71-s − 2.44·73-s − 0.979·79-s + 2.81·83-s − 0.326·85-s − 5.03·89-s + 1.38·95-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.56592\times 10^{12}\) |
| Root analytic conductor: | \(6.18330\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{32} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(11.12086610\) |
| \(L(\frac12)\) | \(\approx\) | \(11.12086610\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 8 T - 214 T^{2} + 3024 T^{3} + 11081 T^{4} - 334096 T^{5} + 380218 T^{6} + 13359336 T^{7} - 18340604 T^{8} + 13359336 p^{3} T^{9} + 380218 p^{6} T^{10} - 334096 p^{9} T^{11} + 11081 p^{12} T^{12} + 3024 p^{15} T^{13} - 214 p^{18} T^{14} - 8 p^{21} T^{15} + p^{24} T^{16} \) |
| 7 | \( 1 - 820 T^{2} + 6912 T^{3} + 7018 p^{2} T^{4} - 4019328 T^{5} - 64496464 T^{6} + 808710912 T^{7} + 10878831475 T^{8} + 808710912 p^{3} T^{9} - 64496464 p^{6} T^{10} - 4019328 p^{9} T^{11} + 7018 p^{14} T^{12} + 6912 p^{15} T^{13} - 820 p^{18} T^{14} + p^{24} T^{16} \) | |
| 11 | \( 1 - 8 T - 3700 T^{2} + 21168 T^{3} + 7424858 T^{4} - 26032312 T^{5} - 10887264464 T^{6} + 1551903240 p T^{7} + 115133848795 p^{2} T^{8} + 1551903240 p^{4} T^{9} - 10887264464 p^{6} T^{10} - 26032312 p^{9} T^{11} + 7424858 p^{12} T^{12} + 21168 p^{15} T^{13} - 3700 p^{18} T^{14} - 8 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 + 4 T - 6498 T^{2} - 1528 p T^{3} + 22134041 T^{4} + 38354040 T^{5} - 68414879138 T^{6} - 25050642164 T^{7} + 179366145158916 T^{8} - 25050642164 p^{3} T^{9} - 68414879138 p^{6} T^{10} + 38354040 p^{9} T^{11} + 22134041 p^{12} T^{12} - 1528 p^{16} T^{13} - 6498 p^{18} T^{14} + 4 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( ( 1 + 16 T + 11870 T^{2} + 166144 T^{3} + 82679203 T^{4} + 166144 p^{3} T^{5} + 11870 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 80 T + 16612 T^{2} - 1115792 T^{3} + 165582646 T^{4} - 1115792 p^{3} T^{5} + 16612 p^{6} T^{6} - 80 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 23 | \( 1 - 200 T + 1436 T^{2} + 4686960 T^{3} - 444057814 T^{4} - 38020548280 T^{5} + 8111534643952 T^{6} + 57304147320 p^{2} T^{7} - 87814317611940365 T^{8} + 57304147320 p^{5} T^{9} + 8111534643952 p^{6} T^{10} - 38020548280 p^{9} T^{11} - 444057814 p^{12} T^{12} + 4686960 p^{15} T^{13} + 1436 p^{18} T^{14} - 200 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 216 T - 46262 T^{2} + 8066160 T^{3} + 2197447561 T^{4} - 182550576240 T^{5} - 81352710116966 T^{6} + 892568522013048 T^{7} + 2580857420026129060 T^{8} + 892568522013048 p^{3} T^{9} - 81352710116966 p^{6} T^{10} - 182550576240 p^{9} T^{11} + 2197447561 p^{12} T^{12} + 8066160 p^{15} T^{13} - 46262 p^{18} T^{14} - 216 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 80 T - 96828 T^{2} - 5357408 T^{3} + 5687064842 T^{4} + 197677265136 T^{5} - 243362680131824 T^{6} - 2307287652097456 T^{7} + 8287488867183853203 T^{8} - 2307287652097456 p^{3} T^{9} - 243362680131824 p^{6} T^{10} + 197677265136 p^{9} T^{11} + 5687064842 p^{12} T^{12} - 5357408 p^{15} T^{13} - 96828 p^{18} T^{14} + 80 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( ( 1 + 276 T + 103330 T^{2} + 18786096 T^{3} + 6284830203 T^{4} + 18786096 p^{3} T^{5} + 103330 p^{6} T^{6} + 276 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 41 | \( 1 - 384 T - 110468 T^{2} + 17468160 T^{3} + 20147453866 T^{4} - 602963473536 T^{5} - 1889776925211152 T^{6} + 76412208039853440 T^{7} + \)\(10\!\cdots\!35\)\( T^{8} + 76412208039853440 p^{3} T^{9} - 1889776925211152 p^{6} T^{10} - 602963473536 p^{9} T^{11} + 20147453866 p^{12} T^{12} + 17468160 p^{15} T^{13} - 110468 p^{18} T^{14} - 384 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 + 160 T - 89508 T^{2} - 54860992 T^{3} + 1213476986 T^{4} + 5154809318880 T^{5} + 1620924566997616 T^{6} - 247583519363785184 T^{7} - \)\(16\!\cdots\!85\)\( T^{8} - 247583519363785184 p^{3} T^{9} + 1620924566997616 p^{6} T^{10} + 5154809318880 p^{9} T^{11} + 1213476986 p^{12} T^{12} - 54860992 p^{15} T^{13} - 89508 p^{18} T^{14} + 160 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 - 768 T + 17188 T^{2} + 40327680 T^{3} + 47344015882 T^{4} - 14617334847744 T^{5} - 4259476291743344 T^{6} - 150592522352063232 T^{7} + \)\(94\!\cdots\!27\)\( T^{8} - 150592522352063232 p^{3} T^{9} - 4259476291743344 p^{6} T^{10} - 14617334847744 p^{9} T^{11} + 47344015882 p^{12} T^{12} + 40327680 p^{15} T^{13} + 17188 p^{18} T^{14} - 768 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( ( 1 + 944 T + 679796 T^{2} + 309404624 T^{3} + 135772939894 T^{4} + 309404624 p^{3} T^{5} + 679796 p^{6} T^{6} + 944 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 59 | \( 1 - 992 T + 132212 T^{2} + 211199808 T^{3} - 64378943398 T^{4} - 30326990708896 T^{5} + 16466029081644112 T^{6} + 1512561630900384 T^{7} - \)\(17\!\cdots\!17\)\( T^{8} + 1512561630900384 p^{3} T^{9} + 16466029081644112 p^{6} T^{10} - 30326990708896 p^{9} T^{11} - 64378943398 p^{12} T^{12} + 211199808 p^{15} T^{13} + 132212 p^{18} T^{14} - 992 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 548 T - 32514 T^{2} + 208018904 T^{3} - 148413039943 T^{4} + 57548939244744 T^{5} - 2541148089226370 T^{6} - 16024239414427665068 T^{7} + \)\(13\!\cdots\!32\)\( T^{8} - 16024239414427665068 p^{3} T^{9} - 2541148089226370 p^{6} T^{10} + 57548939244744 p^{9} T^{11} - 148413039943 p^{12} T^{12} + 208018904 p^{15} T^{13} - 32514 p^{18} T^{14} - 548 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 464 T - 283428 T^{2} + 113886496 T^{3} + 133082038682 T^{4} - 35036923353552 T^{5} + 41403522599668336 T^{6} + 21121861741173813392 T^{7} - \)\(12\!\cdots\!85\)\( T^{8} + 21121861741173813392 p^{3} T^{9} + 41403522599668336 p^{6} T^{10} - 35036923353552 p^{9} T^{11} + 133082038682 p^{12} T^{12} + 113886496 p^{15} T^{13} - 283428 p^{18} T^{14} + 464 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( ( 1 + 1720 T + 2312516 T^{2} + 1956868408 T^{3} + 1389341598886 T^{4} + 1956868408 p^{3} T^{5} + 2312516 p^{6} T^{6} + 1720 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 764 T + 643690 T^{2} - 59411248 T^{3} + 3466474963 T^{4} - 59411248 p^{3} T^{5} + 643690 p^{6} T^{6} + 764 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 79 | \( 1 + 688 T + 263052 T^{2} + 364639328 T^{3} + 78694463402 T^{4} - 134034684327216 T^{5} + 142977502951145392 T^{6} + \)\(13\!\cdots\!84\)\( T^{7} + \)\(39\!\cdots\!03\)\( T^{8} + \)\(13\!\cdots\!84\)\( p^{3} T^{9} + 142977502951145392 p^{6} T^{10} - 134034684327216 p^{9} T^{11} + 78694463402 p^{12} T^{12} + 364639328 p^{15} T^{13} + 263052 p^{18} T^{14} + 688 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 - 2128 T + 1004948 T^{2} + 132639456 T^{3} + 914173696058 T^{4} - 1232388081801584 T^{5} + 303098817229874128 T^{6} - \)\(37\!\cdots\!32\)\( T^{7} + \)\(63\!\cdots\!95\)\( T^{8} - \)\(37\!\cdots\!32\)\( p^{3} T^{9} + 303098817229874128 p^{6} T^{10} - 1232388081801584 p^{9} T^{11} + 914173696058 p^{12} T^{12} + 132639456 p^{15} T^{13} + 1004948 p^{18} T^{14} - 2128 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 + 2112 T + 2710670 T^{2} + 1932604416 T^{3} + 1514938968915 T^{4} + 1932604416 p^{3} T^{5} + 2710670 p^{6} T^{6} + 2112 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( 1 - 1816 T + 720996 T^{2} + 1095214768 T^{3} - 1644010993654 T^{4} + 968182687693848 T^{5} + 8874367889361424 T^{6} - \)\(47\!\cdots\!92\)\( T^{7} + \)\(42\!\cdots\!55\)\( T^{8} - \)\(47\!\cdots\!92\)\( p^{3} T^{9} + 8874367889361424 p^{6} T^{10} + 968182687693848 p^{9} T^{11} - 1644010993654 p^{12} T^{12} + 1095214768 p^{15} T^{13} + 720996 p^{18} T^{14} - 1816 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.19965576940551018622483026281, −4.12743395576972270267858943774, −4.03035380923194546497573252797, −3.46090545493900756364223244751, −3.43753763334217692787362969335, −3.39127329760242773002813399780, −3.20427978439099664126704942335, −3.11165186844255843489050075325, −3.07400714080340161546492485584, −2.87643531316625550844748700859, −2.71379222498787662810446662400, −2.61916878878028927485398013101, −2.39234453044075341383873675516, −2.26760395480274083293187966727, −1.98102269094703141889105041936, −1.79477573936475176740690458952, −1.63755309745304010662441703927, −1.47748251573506453092903554806, −1.20844108707669363024340951387, −1.16184821966741417780784755549, −1.15930015740232643284848692561, −0.67034602112233554670572247768, −0.62574526275503215346349848536, −0.41251172990742218295912755386, −0.17559031081413359921367915925, 0.17559031081413359921367915925, 0.41251172990742218295912755386, 0.62574526275503215346349848536, 0.67034602112233554670572247768, 1.15930015740232643284848692561, 1.16184821966741417780784755549, 1.20844108707669363024340951387, 1.47748251573506453092903554806, 1.63755309745304010662441703927, 1.79477573936475176740690458952, 1.98102269094703141889105041936, 2.26760395480274083293187966727, 2.39234453044075341383873675516, 2.61916878878028927485398013101, 2.71379222498787662810446662400, 2.87643531316625550844748700859, 3.07400714080340161546492485584, 3.11165186844255843489050075325, 3.20427978439099664126704942335, 3.39127329760242773002813399780, 3.43753763334217692787362969335, 3.46090545493900756364223244751, 4.03035380923194546497573252797, 4.12743395576972270267858943774, 4.19965576940551018622483026281
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.