Dirichlet series
| L(s) = 1 | + 48·17-s − 192·19-s + 192·23-s + 88·25-s − 192·29-s − 48·31-s − 256·37-s − 2.01e3·41-s − 224·43-s + 864·47-s − 648·53-s + 336·59-s − 960·61-s − 720·67-s + 2.68e3·71-s − 672·73-s + 1.98e3·79-s − 6.24e3·83-s + 2.16e3·89-s + 4.03e3·97-s + 192·101-s − 2.54e3·103-s − 1.82e3·107-s − 688·109-s + 2.06e3·113-s + 1.18e3·121-s − 1.15e3·125-s + ⋯ |
| L(s) = 1 | + 0.684·17-s − 2.31·19-s + 1.74·23-s + 0.703·25-s − 1.22·29-s − 0.278·31-s − 1.13·37-s − 7.67·41-s − 0.794·43-s + 2.68·47-s − 1.67·53-s + 0.741·59-s − 2.01·61-s − 1.31·67-s + 4.49·71-s − 1.07·73-s + 2.82·79-s − 8.25·83-s + 2.57·89-s + 4.22·97-s + 0.189·101-s − 2.43·103-s − 1.64·107-s − 0.604·109-s + 1.71·113-s + 0.892·121-s − 0.824·125-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.37694\times 10^{16}\) |
| Root analytic conductor: | \(10.2019\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.05386344032\) |
| \(L(\frac12)\) | \(\approx\) | \(0.05386344032\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 88 T^{2} + 1152 T^{3} - 10754 T^{4} - 122688 T^{5} + 1453952 T^{6} + 2017152 T^{7} - 64436621 T^{8} + 2017152 p^{3} T^{9} + 1453952 p^{6} T^{10} - 122688 p^{9} T^{11} - 10754 p^{12} T^{12} + 1152 p^{15} T^{13} - 88 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 108 p T^{2} - 165888 T^{3} + 1058938 T^{4} + 18994176 p T^{5} + 10670277744 T^{6} - 233076787200 T^{7} - 14671410573885 T^{8} - 233076787200 p^{3} T^{9} + 10670277744 p^{6} T^{10} + 18994176 p^{10} T^{11} + 1058938 p^{12} T^{12} - 165888 p^{15} T^{13} - 108 p^{19} T^{14} + p^{24} T^{16} \) | |
| 13 | \( ( 1 + 1696 T^{2} - 31104 T^{3} + 7806642 T^{4} - 31104 p^{3} T^{5} + 1696 p^{6} T^{6} + p^{12} T^{8} )^{2} \) | |
| 17 | \( 1 - 48 T - 7576 T^{2} - 240288 T^{3} + 48058894 T^{4} + 2487877200 T^{5} + 28922298752 T^{6} - 10923088185744 T^{7} - 528788660674109 T^{8} - 10923088185744 p^{3} T^{9} + 28922298752 p^{6} T^{10} + 2487877200 p^{9} T^{11} + 48058894 p^{12} T^{12} - 240288 p^{15} T^{13} - 7576 p^{18} T^{14} - 48 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 192 T + 12708 T^{2} - 163968 T^{3} - 72781094 T^{4} - 532923840 T^{5} + 8300590128 p T^{6} - 46528999732800 T^{7} - 7267852402966365 T^{8} - 46528999732800 p^{3} T^{9} + 8300590128 p^{7} T^{10} - 532923840 p^{9} T^{11} - 72781094 p^{12} T^{12} - 163968 p^{15} T^{13} + 12708 p^{18} T^{14} + 192 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 - 192 T - 684 p T^{2} + 2100864 T^{3} + 651962122 T^{4} - 32825547072 T^{5} - 11082168229968 T^{6} + 212550792485184 T^{7} + 138391974908784435 T^{8} + 212550792485184 p^{3} T^{9} - 11082168229968 p^{6} T^{10} - 32825547072 p^{9} T^{11} + 651962122 p^{12} T^{12} + 2100864 p^{15} T^{13} - 684 p^{19} T^{14} - 192 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( ( 1 + 96 T + 85860 T^{2} + 6375072 T^{3} + 3037151990 T^{4} + 6375072 p^{3} T^{5} + 85860 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 31 | \( 1 + 48 T - 73228 T^{2} + 6457440 T^{3} + 3237603562 T^{4} - 392535047664 T^{5} - 53407701491632 T^{6} + 7268985985847856 T^{7} + 639642386447593363 T^{8} + 7268985985847856 p^{3} T^{9} - 53407701491632 p^{6} T^{10} - 392535047664 p^{9} T^{11} + 3237603562 p^{12} T^{12} + 6457440 p^{15} T^{13} - 73228 p^{18} T^{14} + 48 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 256 T - 1236 T^{2} + 21612032 T^{3} + 4883175962 T^{4} - 343970321664 T^{5} + 366775309137712 T^{6} + 92062738378259200 T^{7} - 1222812831589974909 T^{8} + 92062738378259200 p^{3} T^{9} + 366775309137712 p^{6} T^{10} - 343970321664 p^{9} T^{11} + 4883175962 p^{12} T^{12} + 21612032 p^{15} T^{13} - 1236 p^{18} T^{14} + 256 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( ( 1 + 1008 T + 609784 T^{2} + 250033968 T^{3} + 76165467474 T^{4} + 250033968 p^{3} T^{5} + 609784 p^{6} T^{6} + 1008 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 43 | \( ( 1 + 112 T + 246988 T^{2} + 11722480 T^{3} + 25842449782 T^{4} + 11722480 p^{3} T^{5} + 246988 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 - 864 T + 104276 T^{2} + 10052928 T^{3} + 57768827626 T^{4} - 19567082811168 T^{5} - 2753203620999472 T^{6} - 528511285303912800 T^{7} + \)\(10\!\cdots\!79\)\( T^{8} - 528511285303912800 p^{3} T^{9} - 2753203620999472 p^{6} T^{10} - 19567082811168 p^{9} T^{11} + 57768827626 p^{12} T^{12} + 10052928 p^{15} T^{13} + 104276 p^{18} T^{14} - 864 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 + 648 T + 33012 T^{2} + 21393072 T^{3} + 25022881114 T^{4} - 3843844176456 T^{5} - 250510082721840 T^{6} + 261405578985237720 T^{7} - \)\(45\!\cdots\!65\)\( T^{8} + 261405578985237720 p^{3} T^{9} - 250510082721840 p^{6} T^{10} - 3843844176456 p^{9} T^{11} + 25022881114 p^{12} T^{12} + 21393072 p^{15} T^{13} + 33012 p^{18} T^{14} + 648 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 336 T - 421468 T^{2} + 139166688 T^{3} + 77881695418 T^{4} - 14314611304944 T^{5} - 22878620856915568 T^{6} - 66213470395613328 T^{7} + \)\(66\!\cdots\!91\)\( T^{8} - 66213470395613328 p^{3} T^{9} - 22878620856915568 p^{6} T^{10} - 14314611304944 p^{9} T^{11} + 77881695418 p^{12} T^{12} + 139166688 p^{15} T^{13} - 421468 p^{18} T^{14} - 336 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 960 T + 343872 T^{2} + 246641280 T^{3} + 133370855566 T^{4} + 22982659496640 T^{5} + 19316751371661312 T^{6} + 7509560113650392640 T^{7} - \)\(91\!\cdots\!21\)\( T^{8} + 7509560113650392640 p^{3} T^{9} + 19316751371661312 p^{6} T^{10} + 22982659496640 p^{9} T^{11} + 133370855566 p^{12} T^{12} + 246641280 p^{15} T^{13} + 343872 p^{18} T^{14} + 960 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 720 T - 15340 T^{2} - 253635552 T^{3} - 99254163590 T^{4} + 43727745727728 T^{5} + 46777899615977936 T^{6} + 1218111755932037520 T^{7} - \)\(12\!\cdots\!49\)\( T^{8} + 1218111755932037520 p^{3} T^{9} + 46777899615977936 p^{6} T^{10} + 43727745727728 p^{9} T^{11} - 99254163590 p^{12} T^{12} - 253635552 p^{15} T^{13} - 15340 p^{18} T^{14} + 720 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( ( 1 - 1344 T + 1892084 T^{2} - 1427151168 T^{3} + 1080205217862 T^{4} - 1427151168 p^{3} T^{5} + 1892084 p^{6} T^{6} - 1344 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 73 | \( 1 + 672 T - 637056 T^{2} + 81214656 T^{3} + 513275306398 T^{4} - 128795718341088 T^{5} - 143299066620985344 T^{6} + 22265560577252320992 T^{7} + \)\(15\!\cdots\!59\)\( T^{8} + 22265560577252320992 p^{3} T^{9} - 143299066620985344 p^{6} T^{10} - 128795718341088 p^{9} T^{11} + 513275306398 p^{12} T^{12} + 81214656 p^{15} T^{13} - 637056 p^{18} T^{14} + 672 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 1984 T + 2210820 T^{2} - 2090837888 T^{3} + 1453171977674 T^{4} - 675689441421120 T^{5} + 282155801620181008 T^{6} - 71134456440838937536 T^{7} - \)\(17\!\cdots\!65\)\( T^{8} - 71134456440838937536 p^{3} T^{9} + 282155801620181008 p^{6} T^{10} - 675689441421120 p^{9} T^{11} + 1453171977674 p^{12} T^{12} - 2090837888 p^{15} T^{13} + 2210820 p^{18} T^{14} - 1984 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( ( 1 + 3120 T + 5531020 T^{2} + 6555475248 T^{3} + 5745505983510 T^{4} + 6555475248 p^{3} T^{5} + 5531020 p^{6} T^{6} + 3120 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 - 2160 T + 1322600 T^{2} - 73850400 T^{3} + 48864271150 T^{4} + 257051595430800 T^{5} - 741603917226256000 T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(80\!\cdots\!79\)\( T^{8} + \)\(31\!\cdots\!60\)\( p^{3} T^{9} - 741603917226256000 p^{6} T^{10} + 257051595430800 p^{9} T^{11} + 48864271150 p^{12} T^{12} - 73850400 p^{15} T^{13} + 1322600 p^{18} T^{14} - 2160 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( ( 1 - 2016 T + 2898432 T^{2} - 2840274144 T^{3} + 3044116636418 T^{4} - 2840274144 p^{3} T^{5} + 2898432 p^{6} T^{6} - 2016 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
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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
−3.56110011198134418312394408148, −3.52889291407296043536966168517, −3.22245276208369126449132989448, −3.14146844966943710617663029961, −3.00184894855808617713939352094, −2.89104646644580571035673813878, −2.85530087049187685021999573106, −2.65736271381089980848041455754, −2.62898846559399317047977388266, −2.21214385364944218870295928480, −2.11072814043140903664677507751, −2.01182784723589419450177278172, −1.95165455319357096895911186847, −1.88308029829781666554245958048, −1.67239743317053531787313662944, −1.54998587337137920363213160202, −1.37781871424947191598673515857, −1.31892046860271278619219251326, −1.15068886676920039789260908770, −0.842890353406851890358996873692, −0.800542640501983812840321297512, −0.52230672274947979164136257489, −0.26041146189697737041048973756, −0.25213018043348586144227688714, −0.02245101264375865693927351174, 0.02245101264375865693927351174, 0.25213018043348586144227688714, 0.26041146189697737041048973756, 0.52230672274947979164136257489, 0.800542640501983812840321297512, 0.842890353406851890358996873692, 1.15068886676920039789260908770, 1.31892046860271278619219251326, 1.37781871424947191598673515857, 1.54998587337137920363213160202, 1.67239743317053531787313662944, 1.88308029829781666554245958048, 1.95165455319357096895911186847, 2.01182784723589419450177278172, 2.11072814043140903664677507751, 2.21214385364944218870295928480, 2.62898846559399317047977388266, 2.65736271381089980848041455754, 2.85530087049187685021999573106, 2.89104646644580571035673813878, 3.00184894855808617713939352094, 3.14146844966943710617663029961, 3.22245276208369126449132989448, 3.52889291407296043536966168517, 3.56110011198134418312394408148
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.