Dirichlet series
| L(s) = 1 | + 5-s − 12·7-s + 10·11-s + 9·13-s − 23·17-s − 33·19-s + 31·23-s + 14·25-s + 105·29-s − 12·35-s − 18·41-s − 41·43-s − 107·47-s + 32·49-s − 39·53-s + 10·55-s − 348·59-s − 45·61-s + 9·65-s − 432·67-s + 243·71-s + 16·73-s − 120·77-s + 75·79-s + 82·83-s − 23·85-s + 213·89-s + ⋯ |
| L(s) = 1 | + 1/5·5-s − 1.71·7-s + 0.909·11-s + 9/13·13-s − 1.35·17-s − 1.73·19-s + 1.34·23-s + 0.559·25-s + 3.62·29-s − 0.342·35-s − 0.439·41-s − 0.953·43-s − 2.27·47-s + 0.653·49-s − 0.735·53-s + 2/11·55-s − 5.89·59-s − 0.737·61-s + 9/65·65-s − 6.44·67-s + 3.42·71-s + 0.219·73-s − 1.55·77-s + 0.949·79-s + 0.987·83-s − 0.270·85-s + 2.39·89-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.45589\times 10^{10}\) |
| Root analytic conductor: | \(4.31713\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.004650039527\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004650039527\) |
| \(L(2)\) | 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 | \( 1 \) | |
| 19 | \( 1 + 33 T + 269 T^{2} - 348 p T^{3} - 12042 p T^{4} - 348 p^{3} T^{5} + 269 p^{4} T^{6} + 33 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 5 | \( 1 - T - 13 T^{2} + 292 T^{3} - 421 T^{4} - 5039 T^{5} + 30588 T^{6} + 11733 T^{7} - 851804 T^{8} + 11733 p^{2} T^{9} + 30588 p^{4} T^{10} - 5039 p^{6} T^{11} - 421 p^{8} T^{12} + 292 p^{10} T^{13} - 13 p^{12} T^{14} - p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 6 T + 38 T^{2} + 384 T^{3} + 5862 T^{4} + 384 p^{2} T^{5} + 38 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 11 | \( ( 1 - 5 T + 25 p T^{2} + 186 T^{3} + 32522 T^{4} + 186 p^{2} T^{5} + 25 p^{5} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 13 | \( 1 - 9 T + 337 T^{2} - 2790 T^{3} + 61447 T^{4} - 932715 T^{5} + 10418830 T^{6} - 237377799 T^{7} + 1613988172 T^{8} - 237377799 p^{2} T^{9} + 10418830 p^{4} T^{10} - 932715 p^{6} T^{11} + 61447 p^{8} T^{12} - 2790 p^{10} T^{13} + 337 p^{12} T^{14} - 9 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 23 T - 637 T^{2} - 9464 T^{3} + 442907 T^{4} + 3422479 T^{5} - 10530810 p T^{6} - 371336157 T^{7} + 59307957112 T^{8} - 371336157 p^{2} T^{9} - 10530810 p^{5} T^{10} + 3422479 p^{6} T^{11} + 442907 p^{8} T^{12} - 9464 p^{10} T^{13} - 637 p^{12} T^{14} + 23 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 - 31 T - 589 T^{2} + 27544 T^{3} + 5225 T^{4} - 2594747 T^{5} - 223343094 T^{6} - 2374966005 T^{7} + 280176525940 T^{8} - 2374966005 p^{2} T^{9} - 223343094 p^{4} T^{10} - 2594747 p^{6} T^{11} + 5225 p^{8} T^{12} + 27544 p^{10} T^{13} - 589 p^{12} T^{14} - 31 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 105 T + 7483 T^{2} - 399840 T^{3} + 17830915 T^{4} - 705733587 T^{5} + 25492474348 T^{6} - 842879991579 T^{7} + 25683139528948 T^{8} - 842879991579 p^{2} T^{9} + 25492474348 p^{4} T^{10} - 705733587 p^{6} T^{11} + 17830915 p^{8} T^{12} - 399840 p^{10} T^{13} + 7483 p^{12} T^{14} - 105 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 - 5000 T^{2} + 11932312 T^{4} - 18346108700 T^{6} + 20393419600918 T^{8} - 18346108700 p^{4} T^{10} + 11932312 p^{8} T^{12} - 5000 p^{12} T^{14} + p^{16} T^{16} \) | |
| 37 | \( 1 - 6152 T^{2} + 549244 p T^{4} - 44721393932 T^{6} + 71227113183694 T^{8} - 44721393932 p^{4} T^{10} + 549244 p^{9} T^{12} - 6152 p^{12} T^{14} + p^{16} T^{16} \) | |
| 41 | \( 1 + 18 T + 5284 T^{2} + 93168 T^{3} + 15116683 T^{4} + 203450940 T^{5} + 33524113984 T^{6} + 351143388594 T^{7} + 61749038665600 T^{8} + 351143388594 p^{2} T^{9} + 33524113984 p^{4} T^{10} + 203450940 p^{6} T^{11} + 15116683 p^{8} T^{12} + 93168 p^{10} T^{13} + 5284 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 41 T - 3213 T^{2} - 65936 T^{3} + 7847771 T^{4} + 26461845 T^{5} - 10938362222 T^{6} - 49160999527 T^{7} + 8002416022776 T^{8} - 49160999527 p^{2} T^{9} - 10938362222 p^{4} T^{10} + 26461845 p^{6} T^{11} + 7847771 p^{8} T^{12} - 65936 p^{10} T^{13} - 3213 p^{12} T^{14} + 41 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 107 T + 3773 T^{2} + 4790 p T^{3} + 13157813 T^{4} - 20671799 T^{5} - 13121126406 T^{6} - 814581648117 T^{7} - 68529583926248 T^{8} - 814581648117 p^{2} T^{9} - 13121126406 p^{4} T^{10} - 20671799 p^{6} T^{11} + 13157813 p^{8} T^{12} + 4790 p^{11} T^{13} + 3773 p^{12} T^{14} + 107 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 39 T + 4267 T^{2} + 146640 T^{3} + 123355 T^{4} + 242296731 T^{5} + 18144787570 T^{6} + 2392966790967 T^{7} + 164697791710120 T^{8} + 2392966790967 p^{2} T^{9} + 18144787570 p^{4} T^{10} + 242296731 p^{6} T^{11} + 123355 p^{8} T^{12} + 146640 p^{10} T^{13} + 4267 p^{12} T^{14} + 39 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 348 T + 66940 T^{2} + 9247056 T^{3} + 1010723557 T^{4} + 92075039226 T^{5} + 7241531528608 T^{6} + 502879359972174 T^{7} + 31272462084600580 T^{8} + 502879359972174 p^{2} T^{9} + 7241531528608 p^{4} T^{10} + 92075039226 p^{6} T^{11} + 1010723557 p^{8} T^{12} + 9247056 p^{10} T^{13} + 66940 p^{12} T^{14} + 348 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 45 T - 10313 T^{2} - 428592 T^{3} + 65893339 T^{4} + 2121675759 T^{5} - 290806824308 T^{6} - 3581540902581 T^{7} + 1130193168149644 T^{8} - 3581540902581 p^{2} T^{9} - 290806824308 p^{4} T^{10} + 2121675759 p^{6} T^{11} + 65893339 p^{8} T^{12} - 428592 p^{10} T^{13} - 10313 p^{12} T^{14} + 45 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 432 T + 97276 T^{2} + 15149376 T^{3} + 1809254713 T^{4} + 175861480266 T^{5} + 14614049731768 T^{6} + 1085711859720594 T^{7} + 74881620640414828 T^{8} + 1085711859720594 p^{2} T^{9} + 14614049731768 p^{4} T^{10} + 175861480266 p^{6} T^{11} + 1809254713 p^{8} T^{12} + 15149376 p^{10} T^{13} + 97276 p^{12} T^{14} + 432 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 243 T + 44365 T^{2} - 5997726 T^{3} + 704424943 T^{4} - 70873939629 T^{5} + 6476402096494 T^{6} - 527552062687521 T^{7} + 39543826778862076 T^{8} - 527552062687521 p^{2} T^{9} + 6476402096494 p^{4} T^{10} - 70873939629 p^{6} T^{11} + 704424943 p^{8} T^{12} - 5997726 p^{10} T^{13} + 44365 p^{12} T^{14} - 243 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 - 16 T - 9162 T^{2} + 136456 T^{3} + 9062885 T^{4} + 357581700 T^{5} - 185095090514 T^{6} - 3392958769228 T^{7} + 2623561551973836 T^{8} - 3392958769228 p^{2} T^{9} - 185095090514 p^{4} T^{10} + 357581700 p^{6} T^{11} + 9062885 p^{8} T^{12} + 136456 p^{10} T^{13} - 9162 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 75 T + 13063 T^{2} - 839100 T^{3} + 63496771 T^{4} + 159169725 T^{5} - 185166055982 T^{6} + 52873070751825 T^{7} - 3596786126453096 T^{8} + 52873070751825 p^{2} T^{9} - 185166055982 p^{4} T^{10} + 159169725 p^{6} T^{11} + 63496771 p^{8} T^{12} - 839100 p^{10} T^{13} + 13063 p^{12} T^{14} - 75 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( ( 1 - 41 T + 18101 T^{2} - 1065372 T^{3} + 153581186 T^{4} - 1065372 p^{2} T^{5} + 18101 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 89 | \( 1 - 213 T + 37495 T^{2} - 4765236 T^{3} + 487830907 T^{4} - 48561299361 T^{5} + 4688209891078 T^{6} - 439513920145869 T^{7} + 42578351296229680 T^{8} - 439513920145869 p^{2} T^{9} + 4688209891078 p^{4} T^{10} - 48561299361 p^{6} T^{11} + 487830907 p^{8} T^{12} - 4765236 p^{10} T^{13} + 37495 p^{12} T^{14} - 213 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 144 T + 29986 T^{2} - 3322656 T^{3} + 461958973 T^{4} - 57731126760 T^{5} + 6260707019626 T^{6} - 740261193991848 T^{7} + 64151581335613420 T^{8} - 740261193991848 p^{2} T^{9} + 6260707019626 p^{4} T^{10} - 57731126760 p^{6} T^{11} + 461958973 p^{8} T^{12} - 3322656 p^{10} T^{13} + 29986 p^{12} T^{14} - 144 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−4.49206924025103980242825779612, −4.04452126249383853203867630673, −3.95322740510763020126806403799, −3.78063510670129117129618604190, −3.77006102934894768689115592087, −3.68508683543514630285801726472, −3.59356277569641134185245383218, −3.24233054818578307400433344244, −3.03198007724421813397458871232, −3.01729640888822250729789702503, −2.96554909349977747170315550081, −2.77120337060768305137893653662, −2.59617748309845785073535871021, −2.56391157098206016764141824839, −2.31699853554778535464004370922, −2.08176004261845639375842515134, −1.71514647229517825525324019363, −1.60907196364124311943985958729, −1.57155151910284839020517528730, −1.17471199589354271863438141272, −1.15802739425968110391081189227, −1.14867938524295754643216919005, −0.54587283638538370063533270438, −0.16092004514940260604991598452, −0.01595219982601826374210418592, 0.01595219982601826374210418592, 0.16092004514940260604991598452, 0.54587283638538370063533270438, 1.14867938524295754643216919005, 1.15802739425968110391081189227, 1.17471199589354271863438141272, 1.57155151910284839020517528730, 1.60907196364124311943985958729, 1.71514647229517825525324019363, 2.08176004261845639375842515134, 2.31699853554778535464004370922, 2.56391157098206016764141824839, 2.59617748309845785073535871021, 2.77120337060768305137893653662, 2.96554909349977747170315550081, 3.01729640888822250729789702503, 3.03198007724421813397458871232, 3.24233054818578307400433344244, 3.59356277569641134185245383218, 3.68508683543514630285801726472, 3.77006102934894768689115592087, 3.78063510670129117129618604190, 3.95322740510763020126806403799, 4.04452126249383853203867630673, 4.49206924025103980242825779612
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.