Dirichlet series
| L(s) = 1 | + 8·2-s + 8·3-s + 36·4-s + 64·6-s + 4·7-s + 120·8-s + 31·9-s + 288·12-s − 12·13-s + 32·14-s + 330·16-s − 4·17-s + 248·18-s − 8·19-s + 32·21-s + 14·23-s + 960·24-s − 2·25-s − 96·26-s + 80·27-s + 144·28-s − 2·29-s + 792·32-s − 32·34-s + 1.11e3·36-s + 24·37-s − 64·38-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 4.61·3-s + 18·4-s + 26.1·6-s + 1.51·7-s + 42.4·8-s + 31/3·9-s + 83.1·12-s − 3.32·13-s + 8.55·14-s + 82.5·16-s − 0.970·17-s + 58.4·18-s − 1.83·19-s + 6.98·21-s + 2.91·23-s + 195.·24-s − 2/5·25-s − 18.8·26-s + 15.3·27-s + 27.2·28-s − 0.371·29-s + 140.·32-s − 5.48·34-s + 186·36-s + 3.94·37-s − 10.3·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 11^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.30193\times 10^{12}\) |
| Root analytic conductor: | \(6.05930\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4807.299730\) |
| \(L(\frac12)\) | \(\approx\) | \(4807.299730\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 11 | \( 1 \) | |
| 19 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 - 8 T + 11 p T^{2} - 32 p T^{3} + 77 p T^{4} - 512 T^{5} + 1088 T^{6} - 2168 T^{7} + 3946 T^{8} - 2168 p T^{9} + 1088 p^{2} T^{10} - 512 p^{3} T^{11} + 77 p^{5} T^{12} - 32 p^{6} T^{13} + 11 p^{7} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 2 T^{2} + 12 T^{3} + 6 p T^{4} + 12 T^{5} + 166 T^{6} + 216 T^{7} + 818 T^{8} + 216 p T^{9} + 166 p^{2} T^{10} + 12 p^{3} T^{11} + 6 p^{5} T^{12} + 12 p^{5} T^{13} + 2 p^{6} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 - 4 T + 19 T^{2} - 44 T^{3} + 205 T^{4} - 512 T^{5} + 1982 T^{6} - 4168 T^{7} + 14502 T^{8} - 4168 p T^{9} + 1982 p^{2} T^{10} - 512 p^{3} T^{11} + 205 p^{4} T^{12} - 44 p^{5} T^{13} + 19 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 12 T + 138 T^{2} + 1008 T^{3} + 6933 T^{4} + 37056 T^{5} + 186858 T^{6} + 778068 T^{7} + 3063836 T^{8} + 778068 p T^{9} + 186858 p^{2} T^{10} + 37056 p^{3} T^{11} + 6933 p^{4} T^{12} + 1008 p^{5} T^{13} + 138 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T + 90 T^{2} + 348 T^{3} + 4013 T^{4} + 14472 T^{5} + 115490 T^{6} + 371296 T^{7} + 2324352 T^{8} + 371296 p T^{9} + 115490 p^{2} T^{10} + 14472 p^{3} T^{11} + 4013 p^{4} T^{12} + 348 p^{5} T^{13} + 90 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 14 T + 155 T^{2} - 1274 T^{3} + 9097 T^{4} - 55896 T^{5} + 309446 T^{6} - 1612204 T^{7} + 7857274 T^{8} - 1612204 p T^{9} + 309446 p^{2} T^{10} - 55896 p^{3} T^{11} + 9097 p^{4} T^{12} - 1274 p^{5} T^{13} + 155 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T + 109 T^{2} + 258 T^{3} + 6979 T^{4} + 13888 T^{5} + 311904 T^{6} + 557248 T^{7} + 10254958 T^{8} + 557248 p T^{9} + 311904 p^{2} T^{10} + 13888 p^{3} T^{11} + 6979 p^{4} T^{12} + 258 p^{5} T^{13} + 109 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 116 T^{2} - 368 T^{3} + 6620 T^{4} - 32944 T^{5} + 332140 T^{6} - 1283904 T^{7} + 12924166 T^{8} - 1283904 p T^{9} + 332140 p^{2} T^{10} - 32944 p^{3} T^{11} + 6620 p^{4} T^{12} - 368 p^{5} T^{13} + 116 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 24 T + 430 T^{2} - 5264 T^{3} + 54850 T^{4} - 464736 T^{5} + 3574616 T^{6} - 24087208 T^{7} + 154185111 T^{8} - 24087208 p T^{9} + 3574616 p^{2} T^{10} - 464736 p^{3} T^{11} + 54850 p^{4} T^{12} - 5264 p^{5} T^{13} + 430 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 8 T + 198 T^{2} - 852 T^{3} + 12926 T^{4} - 1068 T^{5} + 277058 T^{6} + 3129640 T^{7} + 898866 T^{8} + 3129640 p T^{9} + 277058 p^{2} T^{10} - 1068 p^{3} T^{11} + 12926 p^{4} T^{12} - 852 p^{5} T^{13} + 198 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 8 T + 248 T^{2} - 1608 T^{3} + 28740 T^{4} - 158760 T^{5} + 2105992 T^{6} - 10010536 T^{7} + 107365334 T^{8} - 10010536 p T^{9} + 2105992 p^{2} T^{10} - 158760 p^{3} T^{11} + 28740 p^{4} T^{12} - 1608 p^{5} T^{13} + 248 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 16 T + 204 T^{2} + 1056 T^{3} + 4438 T^{4} - 28352 T^{5} - 174344 T^{6} - 2301168 T^{7} - 6054281 T^{8} - 2301168 p T^{9} - 174344 p^{2} T^{10} - 28352 p^{3} T^{11} + 4438 p^{4} T^{12} + 1056 p^{5} T^{13} + 204 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 36 T + 837 T^{2} - 14100 T^{3} + 192639 T^{4} - 2217024 T^{5} + 22084800 T^{6} - 193426104 T^{7} + 1494952850 T^{8} - 193426104 p T^{9} + 22084800 p^{2} T^{10} - 2217024 p^{3} T^{11} + 192639 p^{4} T^{12} - 14100 p^{5} T^{13} + 837 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 24 T + 473 T^{2} + 6240 T^{3} + 80511 T^{4} + 846312 T^{5} + 8622712 T^{6} + 73234296 T^{7} + 606983642 T^{8} + 73234296 p T^{9} + 8622712 p^{2} T^{10} + 846312 p^{3} T^{11} + 80511 p^{4} T^{12} + 6240 p^{5} T^{13} + 473 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 12 T + 294 T^{2} - 2352 T^{3} + 30846 T^{4} - 146496 T^{5} + 1421514 T^{6} - 2025300 T^{7} + 49658114 T^{8} - 2025300 p T^{9} + 1421514 p^{2} T^{10} - 146496 p^{3} T^{11} + 30846 p^{4} T^{12} - 2352 p^{5} T^{13} + 294 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 16 T + 410 T^{2} - 4040 T^{3} + 59905 T^{4} - 392832 T^{5} + 4733890 T^{6} - 22797384 T^{7} + 303294756 T^{8} - 22797384 p T^{9} + 4733890 p^{2} T^{10} - 392832 p^{3} T^{11} + 59905 p^{4} T^{12} - 4040 p^{5} T^{13} + 410 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 4 T + 286 T^{2} - 744 T^{3} + 43006 T^{4} - 75896 T^{5} + 4508514 T^{6} - 6235412 T^{7} + 364330690 T^{8} - 6235412 p T^{9} + 4508514 p^{2} T^{10} - 75896 p^{3} T^{11} + 43006 p^{4} T^{12} - 744 p^{5} T^{13} + 286 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 20 T + 314 T^{2} + 3372 T^{3} + 37293 T^{4} + 349560 T^{5} + 3454306 T^{6} + 28436704 T^{7} + 256983872 T^{8} + 28436704 p T^{9} + 3454306 p^{2} T^{10} + 349560 p^{3} T^{11} + 37293 p^{4} T^{12} + 3372 p^{5} T^{13} + 314 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 12 T + 554 T^{2} + 4648 T^{3} + 125246 T^{4} + 753944 T^{5} + 16321006 T^{6} + 950532 p T^{7} + 1483837234 T^{8} + 950532 p^{2} T^{9} + 16321006 p^{2} T^{10} + 753944 p^{3} T^{11} + 125246 p^{4} T^{12} + 4648 p^{5} T^{13} + 554 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 20 T + 430 T^{2} - 5176 T^{3} + 61270 T^{4} - 430904 T^{5} + 3223458 T^{6} - 8701540 T^{7} + 87484498 T^{8} - 8701540 p T^{9} + 3223458 p^{2} T^{10} - 430904 p^{3} T^{11} + 61270 p^{4} T^{12} - 5176 p^{5} T^{13} + 430 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 8 T + 502 T^{2} - 2452 T^{3} + 109054 T^{4} - 285836 T^{5} + 14578098 T^{6} - 18689656 T^{7} + 1449251698 T^{8} - 18689656 p T^{9} + 14578098 p^{2} T^{10} - 285836 p^{3} T^{11} + 109054 p^{4} T^{12} - 2452 p^{5} T^{13} + 502 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 4 T + 324 T^{2} + 180 T^{3} + 38956 T^{4} + 288316 T^{5} + 2398812 T^{6} + 52347924 T^{7} + 144727398 T^{8} + 52347924 p T^{9} + 2398812 p^{2} T^{10} + 288316 p^{3} T^{11} + 38956 p^{4} T^{12} + 180 p^{5} T^{13} + 324 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.48552111041863595975284707363, −3.17726175428099196401104415452, −3.01611749690718161917513252732, −3.00805701484461402141758535237, −2.94399745935233730918627172078, −2.91472850206919615310001540957, −2.87891372068702247118389652247, −2.71770284150304984995287320186, −2.66315133853890061524559869592, −2.55639616365108485053748816480, −2.44999987509506693875100040570, −2.35941968703207989223894749795, −2.19230459402056782753035371882, −2.14280048152680081369307610879, −1.93683475273728521979392440258, −1.92380666777253008022109552665, −1.89038532600464755117081585907, −1.72293873327879412909713685483, −1.61710263310366811897825590658, −1.16994541459959863591064273553, −1.11989537187707316196174697971, −1.02085640991249966557719034849, −0.955214534528607844403087487480, −0.59781427637742864140130726732, −0.19344243548290849982379309018, 0.19344243548290849982379309018, 0.59781427637742864140130726732, 0.955214534528607844403087487480, 1.02085640991249966557719034849, 1.11989537187707316196174697971, 1.16994541459959863591064273553, 1.61710263310366811897825590658, 1.72293873327879412909713685483, 1.89038532600464755117081585907, 1.92380666777253008022109552665, 1.93683475273728521979392440258, 2.14280048152680081369307610879, 2.19230459402056782753035371882, 2.35941968703207989223894749795, 2.44999987509506693875100040570, 2.55639616365108485053748816480, 2.66315133853890061524559869592, 2.71770284150304984995287320186, 2.87891372068702247118389652247, 2.91472850206919615310001540957, 2.94399745935233730918627172078, 3.00805701484461402141758535237, 3.01611749690718161917513252732, 3.17726175428099196401104415452, 3.48552111041863595975284707363
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.