Dirichlet series
| L(s) = 1 | − 128·4-s + 964·7-s + 4.54e3·13-s + 1.02e4·16-s − 2.36e4·19-s + 4.63e4·25-s − 1.23e5·28-s + 7.70e4·31-s − 1.13e4·37-s − 2.26e5·43-s + 6.43e5·49-s − 5.81e5·52-s − 3.27e5·61-s − 6.55e5·64-s + 1.71e6·67-s − 2.18e6·73-s + 3.03e6·76-s − 1.32e6·79-s + 4.37e6·91-s − 2.20e6·97-s − 5.92e6·100-s − 4.49e4·103-s + 4.47e5·109-s + 9.87e6·112-s + 2.06e6·121-s − 9.86e6·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.81·7-s + 2.06·13-s + 5/2·16-s − 3.45·19-s + 2.96·25-s − 5.62·28-s + 2.58·31-s − 0.224·37-s − 2.85·43-s + 5.46·49-s − 4.13·52-s − 1.44·61-s − 5/2·64-s + 5.69·67-s − 5.62·73-s + 6.90·76-s − 2.69·79-s + 5.80·91-s − 2.41·97-s − 5.92·100-s − 0.0411·103-s + 0.345·109-s + 7.02·112-s + 1.16·121-s − 5.17·124-s − 9.70·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.72185\times 10^{12}\) |
| Root analytic conductor: | \(6.10481\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(1.093925034\) |
| \(L(\frac12)\) | \(\approx\) | \(1.093925034\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 46304 T^{2} + 1650266686 T^{4} - 59862115328 p^{4} T^{6} + 1769541248611 p^{8} T^{8} - 59862115328 p^{16} T^{10} + 1650266686 p^{24} T^{12} - 46304 p^{36} T^{14} + p^{48} T^{16} \) |
| 7 | \( ( 1 - 482 T + 26938 T^{2} - 17028398 T^{3} + 18436500226 T^{4} - 17028398 p^{6} T^{5} + 26938 p^{12} T^{6} - 482 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 11 | \( 1 - 2067740 T^{2} + 3021523804504 T^{4} - 2921892925855093364 T^{6} - \)\(30\!\cdots\!26\)\( T^{8} - 2921892925855093364 p^{12} T^{10} + 3021523804504 p^{24} T^{12} - 2067740 p^{36} T^{14} + p^{48} T^{16} \) | |
| 13 | \( ( 1 - 2270 T + 12646702 T^{2} - 30153498176 T^{3} + 78761957653363 T^{4} - 30153498176 p^{6} T^{5} + 12646702 p^{12} T^{6} - 2270 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 17 | \( 1 - 141183800 T^{2} + 9651342187425574 T^{4} - \)\(41\!\cdots\!68\)\( T^{6} + \)\(11\!\cdots\!39\)\( T^{8} - \)\(41\!\cdots\!68\)\( p^{12} T^{10} + 9651342187425574 p^{24} T^{12} - 141183800 p^{36} T^{14} + p^{48} T^{16} \) | |
| 19 | \( ( 1 + 11842 T + 122570434 T^{2} + 382269231070 T^{3} + 2961485052850642 T^{4} + 382269231070 p^{6} T^{5} + 122570434 p^{12} T^{6} + 11842 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 23 | \( 1 - 602193140 T^{2} + 208847216279397736 T^{4} - \)\(21\!\cdots\!20\)\( p T^{6} + \)\(84\!\cdots\!06\)\( T^{8} - \)\(21\!\cdots\!20\)\( p^{13} T^{10} + 208847216279397736 p^{24} T^{12} - 602193140 p^{36} T^{14} + p^{48} T^{16} \) | |
| 29 | \( 1 - 3072361832 T^{2} + 4718062571292062902 T^{4} - \)\(46\!\cdots\!80\)\( T^{6} + \)\(32\!\cdots\!39\)\( T^{8} - \)\(46\!\cdots\!80\)\( p^{12} T^{10} + 4718062571292062902 p^{24} T^{12} - 3072361832 p^{36} T^{14} + p^{48} T^{16} \) | |
| 31 | \( ( 1 - 38528 T + 2054564344 T^{2} - 23224523393216 T^{3} + 1280087508642426862 T^{4} - 23224523393216 p^{6} T^{5} + 2054564344 p^{12} T^{6} - 38528 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 5674 T + 2614216798 T^{2} - 33031702931096 T^{3} + 7074013795851396475 T^{4} - 33031702931096 p^{6} T^{5} + 2614216798 p^{12} T^{6} + 5674 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 41 | \( 1 - 24361730072 T^{2} + \)\(25\!\cdots\!80\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(75\!\cdots\!06\)\( T^{8} - \)\(15\!\cdots\!00\)\( p^{12} T^{10} + \)\(25\!\cdots\!80\)\( p^{24} T^{12} - 24361730072 p^{36} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 + 113302 T + 25372893394 T^{2} + 975408578554 p^{2} T^{3} + \)\(23\!\cdots\!58\)\( T^{4} + 975408578554 p^{8} T^{5} + 25372893394 p^{12} T^{6} + 113302 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( 1 - 45886260248 T^{2} + \)\(10\!\cdots\!48\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{6} + \)\(20\!\cdots\!74\)\( T^{8} - \)\(16\!\cdots\!44\)\( p^{12} T^{10} + \)\(10\!\cdots\!48\)\( p^{24} T^{12} - 45886260248 p^{36} T^{14} + p^{48} T^{16} \) | |
| 53 | \( 1 - 97731632312 T^{2} + \)\(51\!\cdots\!92\)\( T^{4} - \)\(18\!\cdots\!08\)\( T^{6} + \)\(47\!\cdots\!10\)\( T^{8} - \)\(18\!\cdots\!08\)\( p^{12} T^{10} + \)\(51\!\cdots\!92\)\( p^{24} T^{12} - 97731632312 p^{36} T^{14} + p^{48} T^{16} \) | |
| 59 | \( 1 - 68754072824 T^{2} + \)\(28\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{6} + \)\(39\!\cdots\!58\)\( T^{8} - \)\(11\!\cdots\!72\)\( p^{12} T^{10} + \)\(28\!\cdots\!84\)\( p^{24} T^{12} - 68754072824 p^{36} T^{14} + p^{48} T^{16} \) | |
| 61 | \( ( 1 + 163738 T + 5375311858 T^{2} - 12587109571985600 T^{3} - 34867093543223492069 p T^{4} - 12587109571985600 p^{6} T^{5} + 5375311858 p^{12} T^{6} + 163738 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 856646 T + 496161770266 T^{2} - 190187424030542330 T^{3} + \)\(64\!\cdots\!22\)\( T^{4} - 190187424030542330 p^{6} T^{5} + 496161770266 p^{12} T^{6} - 856646 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 71 | \( 1 - 212021007668 T^{2} + \)\(13\!\cdots\!04\)\( T^{4} - \)\(57\!\cdots\!84\)\( T^{6} + \)\(84\!\cdots\!38\)\( T^{8} - \)\(57\!\cdots\!84\)\( p^{12} T^{10} + \)\(13\!\cdots\!04\)\( p^{24} T^{12} - 212021007668 p^{36} T^{14} + p^{48} T^{16} \) | |
| 73 | \( ( 1 + 1094608 T + 746233393222 T^{2} + 336788359861632640 T^{3} + \)\(13\!\cdots\!03\)\( T^{4} + 336788359861632640 p^{6} T^{5} + 746233393222 p^{12} T^{6} + 1094608 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 8398 p T + 858780911746 T^{2} + 463539447756297070 T^{3} + \)\(30\!\cdots\!54\)\( T^{4} + 463539447756297070 p^{6} T^{5} + 858780911746 p^{12} T^{6} + 8398 p^{19} T^{7} + p^{24} T^{8} )^{2} \) | |
| 83 | \( 1 - 1910425105592 T^{2} + \)\(16\!\cdots\!44\)\( T^{4} - \)\(92\!\cdots\!80\)\( T^{6} + \)\(35\!\cdots\!62\)\( T^{8} - \)\(92\!\cdots\!80\)\( p^{12} T^{10} + \)\(16\!\cdots\!44\)\( p^{24} T^{12} - 1910425105592 p^{36} T^{14} + p^{48} T^{16} \) | |
| 89 | \( 1 - 1054207551824 T^{2} + \)\(82\!\cdots\!70\)\( T^{4} - \)\(51\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!75\)\( T^{8} - \)\(51\!\cdots\!84\)\( p^{12} T^{10} + \)\(82\!\cdots\!70\)\( p^{24} T^{12} - 1054207551824 p^{36} T^{14} + p^{48} T^{16} \) | |
| 97 | \( ( 1 + 1100032 T + 2005275894904 T^{2} + 1141210218432366016 T^{3} + \)\(17\!\cdots\!18\)\( T^{4} + 1141210218432366016 p^{6} T^{5} + 2005275894904 p^{12} T^{6} + 1100032 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
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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
−4.60662838388509470817841173791, −4.39816369798921574547822711741, −4.30667027430073679896242956731, −4.30351383789941592818317926919, −4.14551393925702930196530321284, −4.06052551504018568920692979578, −3.69383235514998664572906287737, −3.66519497439467940766105822609, −3.28751230022939231324379122123, −3.12611661831867952698889134618, −2.96089421146667115527579611197, −2.83662686978823647414834873384, −2.48180395293667665606355664628, −2.37315005162079808384891436105, −2.21280817219030915462233519789, −1.74321159128271946207654877190, −1.73262750793074884281652303882, −1.57479375120172598624867436012, −1.19330644621618535124325887074, −1.19162100874893322881330966975, −1.09952539244804774506001404144, −0.873613194864486893900394988954, −0.46212977243488634455341763186, −0.41285624890765573214059246185, −0.07131499567096626553503045981, 0.07131499567096626553503045981, 0.41285624890765573214059246185, 0.46212977243488634455341763186, 0.873613194864486893900394988954, 1.09952539244804774506001404144, 1.19162100874893322881330966975, 1.19330644621618535124325887074, 1.57479375120172598624867436012, 1.73262750793074884281652303882, 1.74321159128271946207654877190, 2.21280817219030915462233519789, 2.37315005162079808384891436105, 2.48180395293667665606355664628, 2.83662686978823647414834873384, 2.96089421146667115527579611197, 3.12611661831867952698889134618, 3.28751230022939231324379122123, 3.66519497439467940766105822609, 3.69383235514998664572906287737, 4.06052551504018568920692979578, 4.14551393925702930196530321284, 4.30351383789941592818317926919, 4.30667027430073679896242956731, 4.39816369798921574547822711741, 4.60662838388509470817841173791
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.