Dirichlet series
| L(s) = 1 | + 29·4-s + 108·5-s − 140·7-s + 2.39e3·9-s − 2.02e3·11-s − 3.95e3·16-s + 6.00e3·17-s + 2.05e4·19-s + 3.13e3·20-s − 5.02e4·23-s − 1.74e4·25-s − 4.06e3·28-s − 1.51e4·35-s + 6.93e4·36-s + 2.60e5·43-s − 5.86e4·44-s + 2.58e5·45-s − 1.00e5·47-s − 6.11e5·49-s − 2.18e5·55-s − 5.45e4·61-s − 3.34e5·63-s − 1.42e5·64-s + 1.74e5·68-s + 4.79e5·73-s + 5.96e5·76-s + 2.83e5·77-s + ⋯ |
| L(s) = 1 | + 0.453·4-s + 0.863·5-s − 0.408·7-s + 3.27·9-s − 1.52·11-s − 0.966·16-s + 1.22·17-s + 2.99·19-s + 0.391·20-s − 4.13·23-s − 1.11·25-s − 0.184·28-s − 0.352·35-s + 1.48·36-s + 3.28·43-s − 0.689·44-s + 2.83·45-s − 0.965·47-s − 5.19·49-s − 1.31·55-s − 0.240·61-s − 1.33·63-s − 0.544·64-s + 0.554·68-s + 1.23·73-s + 1.35·76-s + 0.620·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(133250.\) |
| Root analytic conductor: | \(2.09070\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 19^{8} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(1.678450105\) |
| \(L(\frac12)\) | \(\approx\) | \(1.678450105\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 19 | \( 1 - 20552 T + 7601584 p T^{2} + 14784792 p^{3} T^{3} - 2370602838 p^{5} T^{4} + 14784792 p^{9} T^{5} + 7601584 p^{13} T^{6} - 20552 p^{18} T^{7} + p^{24} T^{8} \) |
| good | 2 | \( 1 - 29 T^{2} + 2399 p T^{4} - 13895 p^{3} T^{6} + 53947 p^{9} T^{8} - 13895 p^{15} T^{10} + 2399 p^{25} T^{12} - 29 p^{36} T^{14} + p^{48} T^{16} \) |
| 3 | \( 1 - 2390 T^{2} + 152507 p^{3} T^{4} - 166426546 p^{3} T^{6} + 144229438508 p^{3} T^{8} - 166426546 p^{15} T^{10} + 152507 p^{27} T^{12} - 2390 p^{36} T^{14} + p^{48} T^{16} \) | |
| 5 | \( ( 1 - 54 T + 2617 p T^{2} + 33438 p^{2} T^{3} + 1701048 p^{3} T^{4} + 33438 p^{8} T^{5} + 2617 p^{13} T^{6} - 54 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 7 | \( ( 1 + 10 p T + 313216 T^{2} - 4775044 T^{3} + 44739584081 T^{4} - 4775044 p^{6} T^{5} + 313216 p^{12} T^{6} + 10 p^{19} T^{7} + p^{24} T^{8} )^{2} \) | |
| 11 | \( ( 1 + 92 p T + 5508247 T^{2} + 4922254784 T^{3} + 13640591783432 T^{4} + 4922254784 p^{6} T^{5} + 5508247 p^{12} T^{6} + 92 p^{19} T^{7} + p^{24} T^{8} )^{2} \) | |
| 13 | \( 1 - 24189110 T^{2} + 279406892674609 T^{4} - \)\(16\!\cdots\!34\)\( p T^{6} + \)\(11\!\cdots\!36\)\( T^{8} - \)\(16\!\cdots\!34\)\( p^{13} T^{10} + 279406892674609 p^{24} T^{12} - 24189110 p^{36} T^{14} + p^{48} T^{16} \) | |
| 17 | \( ( 1 - 3004 T + 69020938 T^{2} - 162546212976 T^{3} + 2144826882821907 T^{4} - 162546212976 p^{6} T^{5} + 69020938 p^{12} T^{6} - 3004 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 23 | \( ( 1 + 25126 T + 535218493 T^{2} + 8162454996578 T^{3} + 107298273301749332 T^{4} + 8162454996578 p^{6} T^{5} + 535218493 p^{12} T^{6} + 25126 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 29 | \( 1 - 1998154190 T^{2} + 1777366869536385073 T^{4} - \)\(10\!\cdots\!58\)\( T^{6} + \)\(69\!\cdots\!48\)\( p^{2} T^{8} - \)\(10\!\cdots\!58\)\( p^{12} T^{10} + 1777366869536385073 p^{24} T^{12} - 1998154190 p^{36} T^{14} + p^{48} T^{16} \) | |
| 31 | \( 1 - 4136753720 T^{2} + 8594799923457707788 T^{4} - \)\(11\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(11\!\cdots\!48\)\( p^{12} T^{10} + 8594799923457707788 p^{24} T^{12} - 4136753720 p^{36} T^{14} + p^{48} T^{16} \) | |
| 37 | \( 1 - 9578108984 T^{2} + 53394864942206427724 T^{4} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(58\!\cdots\!26\)\( T^{8} - \)\(20\!\cdots\!08\)\( p^{12} T^{10} + 53394864942206427724 p^{24} T^{12} - 9578108984 p^{36} T^{14} + p^{48} T^{16} \) | |
| 41 | \( 1 - 375899512 p T^{2} + \)\(10\!\cdots\!24\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{6} + \)\(18\!\cdots\!02\)\( T^{8} - \)\(42\!\cdots\!64\)\( p^{12} T^{10} + \)\(10\!\cdots\!24\)\( p^{24} T^{12} - 375899512 p^{37} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 - 130400 T + 29057706811 T^{2} - 2415257248932016 T^{3} + \)\(28\!\cdots\!16\)\( T^{4} - 2415257248932016 p^{6} T^{5} + 29057706811 p^{12} T^{6} - 130400 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( ( 1 + 50124 T + 25421984975 T^{2} - 29194995849816 T^{3} + \)\(27\!\cdots\!88\)\( T^{4} - 29194995849816 p^{6} T^{5} + 25421984975 p^{12} T^{6} + 50124 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 53 | \( 1 - 129642990110 T^{2} + \)\(80\!\cdots\!29\)\( T^{4} - \)\(31\!\cdots\!02\)\( T^{6} + \)\(83\!\cdots\!76\)\( T^{8} - \)\(31\!\cdots\!02\)\( p^{12} T^{10} + \)\(80\!\cdots\!29\)\( p^{24} T^{12} - 129642990110 p^{36} T^{14} + p^{48} T^{16} \) | |
| 59 | \( 1 - 41207117054 T^{2} + \)\(17\!\cdots\!41\)\( T^{4} - \)\(17\!\cdots\!54\)\( T^{6} - \)\(86\!\cdots\!68\)\( T^{8} - \)\(17\!\cdots\!54\)\( p^{12} T^{10} + \)\(17\!\cdots\!41\)\( p^{24} T^{12} - 41207117054 p^{36} T^{14} + p^{48} T^{16} \) | |
| 61 | \( ( 1 + 27274 T + 84376832509 T^{2} - 5951816481672002 T^{3} + \)\(32\!\cdots\!76\)\( T^{4} - 5951816481672002 p^{6} T^{5} + 84376832509 p^{12} T^{6} + 27274 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 67 | \( 1 - 168095126102 T^{2} + \)\(20\!\cdots\!57\)\( T^{4} - \)\(13\!\cdots\!74\)\( T^{6} + \)\(11\!\cdots\!32\)\( T^{8} - \)\(13\!\cdots\!74\)\( p^{12} T^{10} + \)\(20\!\cdots\!57\)\( p^{24} T^{12} - 168095126102 p^{36} T^{14} + p^{48} T^{16} \) | |
| 71 | \( 1 - 576414235328 T^{2} + \)\(18\!\cdots\!80\)\( T^{4} - \)\(38\!\cdots\!04\)\( T^{6} + \)\(57\!\cdots\!62\)\( T^{8} - \)\(38\!\cdots\!04\)\( p^{12} T^{10} + \)\(18\!\cdots\!80\)\( p^{24} T^{12} - 576414235328 p^{36} T^{14} + p^{48} T^{16} \) | |
| 73 | \( ( 1 - 239984 T + 258827168818 T^{2} + 51452361351736608 T^{3} + \)\(13\!\cdots\!47\)\( T^{4} + 51452361351736608 p^{6} T^{5} + 258827168818 p^{12} T^{6} - 239984 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 79 | \( 1 - 468513052568 T^{2} + \)\(15\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!48\)\( T^{6} + \)\(55\!\cdots\!58\)\( T^{8} - \)\(24\!\cdots\!48\)\( p^{12} T^{10} + \)\(15\!\cdots\!56\)\( p^{24} T^{12} - 468513052568 p^{36} T^{14} + p^{48} T^{16} \) | |
| 83 | \( ( 1 - 241520 T + 986059159792 T^{2} - 240650348200738672 T^{3} + \)\(44\!\cdots\!34\)\( T^{4} - 240650348200738672 p^{6} T^{5} + 986059159792 p^{12} T^{6} - 241520 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 89 | \( 1 - 1943000374496 T^{2} + \)\(13\!\cdots\!64\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{6} - \)\(20\!\cdots\!42\)\( T^{8} - \)\(29\!\cdots\!08\)\( p^{12} T^{10} + \)\(13\!\cdots\!64\)\( p^{24} T^{12} - 1943000374496 p^{36} T^{14} + p^{48} T^{16} \) | |
| 97 | \( 1 - 2023436359424 T^{2} + \)\(15\!\cdots\!16\)\( T^{4} - \)\(94\!\cdots\!24\)\( T^{6} + \)\(82\!\cdots\!90\)\( T^{8} - \)\(94\!\cdots\!24\)\( p^{12} T^{10} + \)\(15\!\cdots\!16\)\( p^{24} T^{12} - 2023436359424 p^{36} T^{14} + p^{48} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76888576978965309265911595912, −7.59504244092420670711554032230, −7.46141468750512795208783108068, −7.41328193950356372890368011949, −6.78731464363435454739132857359, −6.72703384211441407156213490463, −6.45351999736207093452659194072, −6.33798897140307399827236074430, −5.80111195165395356311207723466, −5.62576795645179455268057380603, −5.61826905185588056994307192375, −5.23946804703489896043285329222, −4.80786852122013080165790444538, −4.42177070626662322550645921197, −4.39395044549187847461462632202, −4.09188953663521930284471797219, −3.54878702769351381433696372813, −3.29611306239370129424488288852, −3.09042890455584303974679210582, −2.32000696575438243898152604069, −1.98851904663012117812876644586, −1.89969318899710707443652956283, −1.31533835104711982211965979643, −1.08239387524706746849662190502, −0.18485158753735972823851866807, 0.18485158753735972823851866807, 1.08239387524706746849662190502, 1.31533835104711982211965979643, 1.89969318899710707443652956283, 1.98851904663012117812876644586, 2.32000696575438243898152604069, 3.09042890455584303974679210582, 3.29611306239370129424488288852, 3.54878702769351381433696372813, 4.09188953663521930284471797219, 4.39395044549187847461462632202, 4.42177070626662322550645921197, 4.80786852122013080165790444538, 5.23946804703489896043285329222, 5.61826905185588056994307192375, 5.62576795645179455268057380603, 5.80111195165395356311207723466, 6.33798897140307399827236074430, 6.45351999736207093452659194072, 6.72703384211441407156213490463, 6.78731464363435454739132857359, 7.41328193950356372890368011949, 7.46141468750512795208783108068, 7.59504244092420670711554032230, 7.76888576978965309265911595912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.