Dirichlet series
| L(s) = 1 | + 12·3-s − 6·5-s + 8·7-s + 78·9-s − 22·11-s − 72·15-s + 36·17-s + 42·19-s + 96·21-s + 48·23-s − 11·25-s + 360·27-s + 68·29-s − 60·31-s − 264·33-s − 48·35-s − 118·37-s − 92·43-s − 468·45-s − 12·47-s + 22·49-s + 432·51-s + 10·53-s + 132·55-s + 504·57-s − 54·59-s + 24·61-s + ⋯ |
| L(s) = 1 | + 4·3-s − 6/5·5-s + 8/7·7-s + 26/3·9-s − 2·11-s − 4.79·15-s + 2.11·17-s + 2.21·19-s + 32/7·21-s + 2.08·23-s − 0.439·25-s + 40/3·27-s + 2.34·29-s − 1.93·31-s − 8·33-s − 1.37·35-s − 3.18·37-s − 2.13·43-s − 10.3·45-s − 0.255·47-s + 0.448·49-s + 8.47·51-s + 0.188·53-s + 12/5·55-s + 8.84·57-s − 0.915·59-s + 0.393·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(192821.\) |
| Root analytic conductor: | \(2.13954\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(26.16618086\) |
| \(L(\frac12)\) | \(\approx\) | \(26.16618086\) |
| \(L(2)\) | 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 | \( ( 1 - p T + p T^{2} )^{4} \) | |
| 7 | \( 1 - 8 T + 6 p T^{2} - 16 p T^{3} - 37 p^{2} T^{4} - 16 p^{3} T^{5} + 6 p^{5} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 5 | \( 1 + 6 T + 47 T^{2} + 42 p T^{3} + 261 T^{4} + 492 p T^{5} + 15082 T^{6} + 133176 T^{7} + 1092986 T^{8} + 133176 p^{2} T^{9} + 15082 p^{4} T^{10} + 492 p^{7} T^{11} + 261 p^{8} T^{12} + 42 p^{11} T^{13} + 47 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 + 2 p T + 43 T^{2} - 3622 T^{3} - 33751 T^{4} + 150652 T^{5} + 3545774 T^{6} + 14111576 T^{7} - 47473454 T^{8} + 14111576 p^{2} T^{9} + 3545774 p^{4} T^{10} + 150652 p^{6} T^{11} - 33751 p^{8} T^{12} - 3622 p^{10} T^{13} + 43 p^{12} T^{14} + 2 p^{15} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 1090 T^{2} + 553857 T^{4} - 171023522 T^{6} + 35061921764 T^{8} - 171023522 p^{4} T^{10} + 553857 p^{8} T^{12} - 1090 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 - 36 T + 1128 T^{2} - 25056 T^{3} + 423426 T^{4} - 3912084 T^{5} + 3929664 T^{6} + 1266732252 T^{7} - 28295557165 T^{8} + 1266732252 p^{2} T^{9} + 3929664 p^{4} T^{10} - 3912084 p^{6} T^{11} + 423426 p^{8} T^{12} - 25056 p^{10} T^{13} + 1128 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 42 T + 1487 T^{2} - 37758 T^{3} + 939597 T^{4} - 19816692 T^{5} + 377239378 T^{6} - 7257982200 T^{7} + 133297511354 T^{8} - 7257982200 p^{2} T^{9} + 377239378 p^{4} T^{10} - 19816692 p^{6} T^{11} + 939597 p^{8} T^{12} - 37758 p^{10} T^{13} + 1487 p^{12} T^{14} - 42 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 - 48 T + 1092 T^{2} + 3936 T^{3} - 894758 T^{4} + 29740848 T^{5} - 203857200 T^{6} - 10675090896 T^{7} + 444713730579 T^{8} - 10675090896 p^{2} T^{9} - 203857200 p^{4} T^{10} + 29740848 p^{6} T^{11} - 894758 p^{8} T^{12} + 3936 p^{10} T^{13} + 1092 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( ( 1 - 34 T + 2301 T^{2} - 56918 T^{3} + 2679848 T^{4} - 56918 p^{2} T^{5} + 2301 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 60 T + 3950 T^{2} + 165000 T^{3} + 6447417 T^{4} + 193426680 T^{5} + 5752291150 T^{6} + 148730276100 T^{7} + 4519543033748 T^{8} + 148730276100 p^{2} T^{9} + 5752291150 p^{4} T^{10} + 193426680 p^{6} T^{11} + 6447417 p^{8} T^{12} + 165000 p^{10} T^{13} + 3950 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 118 T + 4731 T^{2} + 79034 T^{3} + 3881129 T^{4} + 286528380 T^{5} + 10369543726 T^{6} + 385177185496 T^{7} + 16686145262322 T^{8} + 385177185496 p^{2} T^{9} + 10369543726 p^{4} T^{10} + 286528380 p^{6} T^{11} + 3881129 p^{8} T^{12} + 79034 p^{10} T^{13} + 4731 p^{12} T^{14} + 118 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 6168 T^{2} + 21108060 T^{4} - 52972041384 T^{6} + 102006783313478 T^{8} - 52972041384 p^{4} T^{10} + 21108060 p^{8} T^{12} - 6168 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( ( 1 + 46 T + 3673 T^{2} + 131086 T^{3} + 8403460 T^{4} + 131086 p^{2} T^{5} + 3673 p^{4} T^{6} + 46 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 47 | \( 1 + 12 T + 6776 T^{2} + 80736 T^{3} + 26141058 T^{4} + 690376668 T^{5} + 73552782592 T^{6} + 2491793821644 T^{7} + 160966087977299 T^{8} + 2491793821644 p^{2} T^{9} + 73552782592 p^{4} T^{10} + 690376668 p^{6} T^{11} + 26141058 p^{8} T^{12} + 80736 p^{10} T^{13} + 6776 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 10 T - 5201 T^{2} + 194170 T^{3} + 11643389 T^{4} - 697888660 T^{5} + 14769549650 T^{6} + 1159516055440 T^{7} - 90771774536198 T^{8} + 1159516055440 p^{2} T^{9} + 14769549650 p^{4} T^{10} - 697888660 p^{6} T^{11} + 11643389 p^{8} T^{12} + 194170 p^{10} T^{13} - 5201 p^{12} T^{14} - 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 54 T + 9043 T^{2} + 435834 T^{3} + 41825857 T^{4} + 2464471980 T^{5} + 133033982470 T^{6} + 10148662706472 T^{7} + 404005950748642 T^{8} + 10148662706472 p^{2} T^{9} + 133033982470 p^{4} T^{10} + 2464471980 p^{6} T^{11} + 41825857 p^{8} T^{12} + 435834 p^{10} T^{13} + 9043 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 24 T + 2068 T^{2} - 45024 T^{3} + 7099018 T^{4} - 814973400 T^{5} + 12893636176 T^{6} - 1282247801592 T^{7} + 32390202015283 T^{8} - 1282247801592 p^{2} T^{9} + 12893636176 p^{4} T^{10} - 814973400 p^{6} T^{11} + 7099018 p^{8} T^{12} - 45024 p^{10} T^{13} + 2068 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 22 T - 10073 T^{2} - 243818 T^{3} + 56790101 T^{4} + 2497639460 T^{5} - 152314855990 T^{6} - 6901504668560 T^{7} + 437152448364490 T^{8} - 6901504668560 p^{2} T^{9} - 152314855990 p^{4} T^{10} + 2497639460 p^{6} T^{11} + 56790101 p^{8} T^{12} - 243818 p^{10} T^{13} - 10073 p^{12} T^{14} - 22 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 196 T + 22624 T^{2} + 2070604 T^{3} + 164062462 T^{4} + 2070604 p^{2} T^{5} + 22624 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 138 T + 15867 T^{2} + 1313622 T^{3} + 94988169 T^{4} + 6738952980 T^{5} + 589937952558 T^{6} + 53763061993224 T^{7} + 4139076271929602 T^{8} + 53763061993224 p^{2} T^{9} + 589937952558 p^{4} T^{10} + 6738952980 p^{6} T^{11} + 94988169 p^{8} T^{12} + 1313622 p^{10} T^{13} + 15867 p^{12} T^{14} + 138 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 164 T + 16510 T^{2} - 3029320 T^{3} + 328280393 T^{4} - 25606577864 T^{5} + 3087143881598 T^{6} - 266204700833164 T^{7} + 17086123801230196 T^{8} - 266204700833164 p^{2} T^{9} + 3087143881598 p^{4} T^{10} - 25606577864 p^{6} T^{11} + 328280393 p^{8} T^{12} - 3029320 p^{10} T^{13} + 16510 p^{12} T^{14} - 164 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 50682 T^{2} + 1152460209 T^{4} - 15336635334306 T^{6} + 130556747398908548 T^{8} - 15336635334306 p^{4} T^{10} + 1152460209 p^{8} T^{12} - 50682 p^{12} T^{14} + p^{16} T^{16} \) | |
| 89 | \( 1 + 60 T + 25008 T^{2} + 1428480 T^{3} + 347709714 T^{4} + 26548130220 T^{5} + 3690840157344 T^{6} + 302551794909660 T^{7} + 30452609075742995 T^{8} + 302551794909660 p^{2} T^{9} + 3690840157344 p^{4} T^{10} + 26548130220 p^{6} T^{11} + 347709714 p^{8} T^{12} + 1428480 p^{10} T^{13} + 25008 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 10002 T^{2} + 257682129 T^{4} - 2790925375890 T^{6} + 29923705256218532 T^{8} - 2790925375890 p^{4} T^{10} + 257682129 p^{8} T^{12} - 10002 p^{12} T^{14} + p^{16} T^{16} \) | |
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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
−5.61761423471907722103970384657, −5.20705838262580329216292294300, −5.14887514539449677988197920933, −5.06161877075322594920731885065, −4.80860699404692409846444192450, −4.75815182553152690742844016884, −4.70665858153162349538399134469, −4.41772902568040877876977336752, −4.10922078065697510283257994864, −3.94584325192606163685167029269, −3.76277017503891866087337415519, −3.43591083709715514662034647157, −3.30511412115956214605146424378, −3.16860732095385184147240417813, −3.16087033040613091671352190782, −3.04784191178926785480489317853, −3.02992347569008823257908954059, −2.61153502811589543929399640780, −2.33106663346352155169963918827, −1.86577958900276500586316840339, −1.79073399981075546022736915367, −1.64084189549961924748307574341, −1.49569905305426011655270356727, −0.78608874563229262953639382606, −0.56211963567798624412200170884, 0.56211963567798624412200170884, 0.78608874563229262953639382606, 1.49569905305426011655270356727, 1.64084189549961924748307574341, 1.79073399981075546022736915367, 1.86577958900276500586316840339, 2.33106663346352155169963918827, 2.61153502811589543929399640780, 3.02992347569008823257908954059, 3.04784191178926785480489317853, 3.16087033040613091671352190782, 3.16860732095385184147240417813, 3.30511412115956214605146424378, 3.43591083709715514662034647157, 3.76277017503891866087337415519, 3.94584325192606163685167029269, 4.10922078065697510283257994864, 4.41772902568040877876977336752, 4.70665858153162349538399134469, 4.75815182553152690742844016884, 4.80860699404692409846444192450, 5.06161877075322594920731885065, 5.14887514539449677988197920933, 5.20705838262580329216292294300, 5.61761423471907722103970384657
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.