Dirichlet series
| L(s) = 1 | − 2·2-s + 2·4-s − 4·7-s + 10·8-s + 6·11-s + 8·14-s − 18·16-s − 12·17-s − 16·19-s − 12·22-s + 152·25-s − 8·28-s − 128·29-s − 10·31-s + 2·32-s + 24·34-s − 84·37-s + 32·38-s − 20·41-s − 190·43-s + 12·44-s − 58·47-s − 224·49-s − 304·50-s − 252·53-s − 40·56-s + 256·58-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 4/7·7-s + 5/4·8-s + 6/11·11-s + 4/7·14-s − 9/8·16-s − 0.705·17-s − 0.842·19-s − 0.545·22-s + 6.07·25-s − 2/7·28-s − 4.41·29-s − 0.322·31-s + 1/16·32-s + 0.705·34-s − 2.27·37-s + 0.842·38-s − 0.487·41-s − 4.41·43-s + 3/11·44-s − 1.23·47-s − 4.57·49-s − 6.07·50-s − 4.75·53-s − 5/7·56-s + 4.41·58-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 29^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.54341\times 10^{6}\) |
| Root analytic conductor: | \(2.66678\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 29^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.0006556672414\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0006556672414\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 128 T + 7400 T^{2} + 320 p^{2} T^{3} + 9518 p^{2} T^{4} + 320 p^{4} T^{5} + 7400 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 2 | \( 1 + p T + p T^{2} - 5 p T^{3} - 13 p T^{4} - 9 p T^{5} + 33 p T^{6} + 45 p T^{7} + 321 T^{8} + 45 p^{3} T^{9} + 33 p^{5} T^{10} - 9 p^{7} T^{11} - 13 p^{9} T^{12} - 5 p^{11} T^{13} + p^{13} T^{14} + p^{15} T^{15} + p^{16} T^{16} \) |
| 5 | \( 1 - 152 T^{2} + 11042 T^{4} - 495504 T^{6} + 14942291 T^{8} - 495504 p^{4} T^{10} + 11042 p^{8} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \) | |
| 7 | \( ( 1 + 2 T + 118 T^{2} + 262 T^{3} + 6782 T^{4} + 262 p^{2} T^{5} + 118 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 11 | \( 1 - 6 T + 18 T^{2} + 984 T^{3} + 5458 T^{4} - 25326 T^{5} + 537840 T^{6} + 21947202 T^{7} - 268164717 T^{8} + 21947202 p^{2} T^{9} + 537840 p^{4} T^{10} - 25326 p^{6} T^{11} + 5458 p^{8} T^{12} + 984 p^{10} T^{13} + 18 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 900 T^{2} + 416090 T^{4} - 121830768 T^{6} + 24605646387 T^{8} - 121830768 p^{4} T^{10} + 416090 p^{8} T^{12} - 900 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 + 12 T + 72 T^{2} + 156 T^{3} - 11400 T^{4} + 788916 T^{5} + 35640 p^{2} T^{6} + 16260516 p T^{7} + 7918453918 T^{8} + 16260516 p^{3} T^{9} + 35640 p^{6} T^{10} + 788916 p^{6} T^{11} - 11400 p^{8} T^{12} + 156 p^{10} T^{13} + 72 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 16 T + 128 T^{2} - 9968 T^{3} - 125288 T^{4} + 124848 T^{5} + 67714944 T^{6} - 106114512 T^{7} + 873564318 T^{8} - 106114512 p^{2} T^{9} + 67714944 p^{4} T^{10} + 124848 p^{6} T^{11} - 125288 p^{8} T^{12} - 9968 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( ( 1 + 1798 T^{2} + 204 T^{3} + 1362542 T^{4} + 204 p^{2} T^{5} + 1798 p^{4} T^{6} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 10 T + 50 T^{2} - 18752 T^{3} - 59126 T^{4} + 5169114 T^{5} + 230466192 T^{6} - 14955517326 T^{7} - 982612297341 T^{8} - 14955517326 p^{2} T^{9} + 230466192 p^{4} T^{10} + 5169114 p^{6} T^{11} - 59126 p^{8} T^{12} - 18752 p^{10} T^{13} + 50 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 + 84 T + 3528 T^{2} + 71652 T^{3} - 2089980 T^{4} - 123205908 T^{5} - 408842280 T^{6} + 218820380604 T^{7} + 13702947378118 T^{8} + 218820380604 p^{2} T^{9} - 408842280 p^{4} T^{10} - 123205908 p^{6} T^{11} - 2089980 p^{8} T^{12} + 71652 p^{10} T^{13} + 3528 p^{12} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 + 20 T + 200 T^{2} + 65948 T^{3} + 2077432 T^{4} - 115380012 T^{5} - 548517288 T^{6} - 112249030692 T^{7} - 9153258882402 T^{8} - 112249030692 p^{2} T^{9} - 548517288 p^{4} T^{10} - 115380012 p^{6} T^{11} + 2077432 p^{8} T^{12} + 65948 p^{10} T^{13} + 200 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 190 T + 18050 T^{2} + 1310896 T^{3} + 89172690 T^{4} + 5513705486 T^{5} + 297261149248 T^{6} + 14457675945470 T^{7} + 648656884794643 T^{8} + 14457675945470 p^{2} T^{9} + 297261149248 p^{4} T^{10} + 5513705486 p^{6} T^{11} + 89172690 p^{8} T^{12} + 1310896 p^{10} T^{13} + 18050 p^{12} T^{14} + 190 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 58 T + 1682 T^{2} + 75904 T^{3} + 8447626 T^{4} + 469112154 T^{5} + 7188912 p^{2} T^{6} + 15968051454 p T^{7} + 30625787988195 T^{8} + 15968051454 p^{3} T^{9} + 7188912 p^{6} T^{10} + 469112154 p^{6} T^{11} + 8447626 p^{8} T^{12} + 75904 p^{10} T^{13} + 1682 p^{12} T^{14} + 58 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 + 126 T + 12120 T^{2} + 805920 T^{3} + 48326113 T^{4} + 805920 p^{2} T^{5} + 12120 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( ( 1 - 20 T + 4038 T^{2} + 142432 T^{3} + 879134 T^{4} + 142432 p^{2} T^{5} + 4038 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 61 | \( 1 + 208 T + 21632 T^{2} + 2017096 T^{3} + 184647288 T^{4} + 13664724728 T^{5} + 882310746016 T^{6} + 58889137003472 T^{7} + 3818097286447198 T^{8} + 58889137003472 p^{2} T^{9} + 882310746016 p^{4} T^{10} + 13664724728 p^{6} T^{11} + 184647288 p^{8} T^{12} + 2017096 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 11008 T^{2} + 27285916 T^{4} + 201250654976 T^{6} - 1694675724118778 T^{8} + 201250654976 p^{4} T^{10} + 27285916 p^{8} T^{12} - 11008 p^{12} T^{14} + p^{16} T^{16} \) | |
| 71 | \( 1 - 21324 T^{2} + 250110380 T^{4} - 1967023408740 T^{6} + 11455271890903830 T^{8} - 1967023408740 p^{4} T^{10} + 250110380 p^{8} T^{12} - 21324 p^{12} T^{14} + p^{16} T^{16} \) | |
| 73 | \( 1 + 188 T + 17672 T^{2} + 1559180 T^{3} + 182719716 T^{4} + 18486242500 T^{5} + 1461911905048 T^{6} + 113727191207188 T^{7} + 8685700663672390 T^{8} + 113727191207188 p^{2} T^{9} + 1461911905048 p^{4} T^{10} + 18486242500 p^{6} T^{11} + 182719716 p^{8} T^{12} + 1559180 p^{10} T^{13} + 17672 p^{12} T^{14} + 188 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 + 382 T + 72962 T^{2} + 10210992 T^{3} + 1201614634 T^{4} + 122408323406 T^{5} + 11219951427216 T^{6} + 967189248125926 T^{7} + 78871519191926211 T^{8} + 967189248125926 p^{2} T^{9} + 11219951427216 p^{4} T^{10} + 122408323406 p^{6} T^{11} + 1201614634 p^{8} T^{12} + 10210992 p^{10} T^{13} + 72962 p^{12} T^{14} + 382 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( ( 1 + 140 T + 22718 T^{2} + 2108544 T^{3} + 207732422 T^{4} + 2108544 p^{2} T^{5} + 22718 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 89 | \( 1 - 64 T + 2048 T^{2} + 960272 T^{3} - 17872712 T^{4} + 244915632 T^{5} + 481989870720 T^{6} + 41121933793344 T^{7} - 1442843088822882 T^{8} + 41121933793344 p^{2} T^{9} + 481989870720 p^{4} T^{10} + 244915632 p^{6} T^{11} - 17872712 p^{8} T^{12} + 960272 p^{10} T^{13} + 2048 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 44 T + 968 T^{2} - 45508 T^{3} + 12434104 T^{4} + 5439303060 T^{5} + 228328611000 T^{6} + 46830606041796 T^{7} + 9294107220541086 T^{8} + 46830606041796 p^{2} T^{9} + 228328611000 p^{4} T^{10} + 5439303060 p^{6} T^{11} + 12434104 p^{8} T^{12} - 45508 p^{10} T^{13} + 968 p^{12} T^{14} + 44 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−4.98747226650138550616640252995, −4.91711141613747991923835909870, −4.87030762679645328382984741743, −4.73288205364098891311227694680, −4.66628235395561141342551207421, −4.39751182409181706407391113368, −4.37838499300261392925361758752, −3.99062497481418541022850324919, −3.84941048247746449105375495154, −3.59267828356608614895019747818, −3.34066383667063266520288395341, −3.18569580329704350859517345341, −3.06213542384532207246845412158, −3.02459212502515606869745690176, −3.01500562697819009503878802877, −2.92557861774182032329452598611, −2.06411102709438302984418784879, −1.81121471379935329231339259923, −1.81039114868261209579658761965, −1.67011014177566652360565993152, −1.62080304477832414013047250442, −1.23314770156362503039558082347, −1.19516676160692351720354357401, −0.04685008673065721138079459684, −0.04168614100663665924087266716, 0.04168614100663665924087266716, 0.04685008673065721138079459684, 1.19516676160692351720354357401, 1.23314770156362503039558082347, 1.62080304477832414013047250442, 1.67011014177566652360565993152, 1.81039114868261209579658761965, 1.81121471379935329231339259923, 2.06411102709438302984418784879, 2.92557861774182032329452598611, 3.01500562697819009503878802877, 3.02459212502515606869745690176, 3.06213542384532207246845412158, 3.18569580329704350859517345341, 3.34066383667063266520288395341, 3.59267828356608614895019747818, 3.84941048247746449105375495154, 3.99062497481418541022850324919, 4.37838499300261392925361758752, 4.39751182409181706407391113368, 4.66628235395561141342551207421, 4.73288205364098891311227694680, 4.87030762679645328382984741743, 4.91711141613747991923835909870, 4.98747226650138550616640252995
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.