Dirichlet series
| L(s) = 1 | − 11·4-s + 24·11-s + 72·16-s + 8·19-s − 114·29-s + 28·31-s − 102·41-s − 264·44-s + 265·49-s + 120·59-s − 50·61-s − 337·64-s − 88·76-s + 128·79-s + 816·101-s − 52·109-s + 1.25e3·116-s − 364·121-s − 308·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2.75·4-s + 2.18·11-s + 9/2·16-s + 8/19·19-s − 3.93·29-s + 0.903·31-s − 2.48·41-s − 6·44-s + 5.40·49-s + 2.03·59-s − 0.819·61-s − 5.26·64-s − 1.15·76-s + 1.62·79-s + 8.07·101-s − 0.477·109-s + 10.8·116-s − 3.00·121-s − 2.48·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{60} \cdot 5^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.96235\times 10^{25}\) |
| Root analytic conductor: | \(4.28863\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{60} \cdot 5^{40} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.2159118787\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2159118787\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + 11 T^{2} + 49 T^{4} + 21 p^{2} T^{6} - 177 T^{8} - 1035 T^{10} + 3375 T^{12} + 19995 p T^{14} + 125067 T^{16} + 145709 T^{18} + 48577 T^{20} + 145709 p^{4} T^{22} + 125067 p^{8} T^{24} + 19995 p^{13} T^{26} + 3375 p^{16} T^{28} - 1035 p^{20} T^{30} - 177 p^{24} T^{32} + 21 p^{30} T^{34} + 49 p^{32} T^{36} + 11 p^{36} T^{38} + p^{40} T^{40} \) |
| 7 | \( 1 - 265 T^{2} + 33055 T^{4} - 2729172 T^{6} + 183594576 T^{8} - 11541113544 T^{10} + 706999492509 T^{12} - 41608324291335 T^{14} + 2276847521121615 T^{16} - 114666352040929084 T^{18} + 5589021393801754588 T^{20} - 114666352040929084 p^{4} T^{22} + 2276847521121615 p^{8} T^{24} - 41608324291335 p^{12} T^{26} + 706999492509 p^{16} T^{28} - 11541113544 p^{20} T^{30} + 183594576 p^{24} T^{32} - 2729172 p^{28} T^{34} + 33055 p^{32} T^{36} - 265 p^{36} T^{38} + p^{40} T^{40} \) | |
| 11 | \( ( 1 - 12 T + 398 T^{2} - 4200 T^{3} + 78909 T^{4} - 366018 T^{5} + 5983320 T^{6} + 58257912 T^{7} - 572658579 T^{8} + 21513536958 T^{9} - 151209444198 T^{10} + 21513536958 p^{2} T^{11} - 572658579 p^{4} T^{12} + 58257912 p^{6} T^{13} + 5983320 p^{8} T^{14} - 366018 p^{10} T^{15} + 78909 p^{12} T^{16} - 4200 p^{14} T^{17} + 398 p^{16} T^{18} - 12 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 13 | \( 1 - 76 p T^{2} + 37171 p T^{4} - 158139444 T^{6} + 3073345095 p T^{8} - 8648465378916 T^{10} + 1718569437415782 T^{12} - 24579537564713748 p T^{14} + 55174452755456899653 T^{16} - \)\(69\!\cdots\!08\)\( p T^{18} + \)\(14\!\cdots\!69\)\( T^{20} - \)\(69\!\cdots\!08\)\( p^{5} T^{22} + 55174452755456899653 p^{8} T^{24} - 24579537564713748 p^{13} T^{26} + 1718569437415782 p^{16} T^{28} - 8648465378916 p^{20} T^{30} + 3073345095 p^{25} T^{32} - 158139444 p^{28} T^{34} + 37171 p^{33} T^{36} - 76 p^{37} T^{38} + p^{40} T^{40} \) | |
| 17 | \( ( 1 - 2186 T^{2} + 2296254 T^{4} - 1522812012 T^{6} + 704436708009 T^{8} - 237175276026432 T^{10} + 704436708009 p^{4} T^{12} - 1522812012 p^{8} T^{14} + 2296254 p^{12} T^{16} - 2186 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 19 | \( ( 1 - 2 T + 1230 T^{2} - 4716 T^{3} + 694653 T^{4} - 2875722 T^{5} + 694653 p^{2} T^{6} - 4716 p^{4} T^{7} + 1230 p^{6} T^{8} - 2 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 23 | \( 1 + 4271 T^{2} + 9699319 T^{4} + 15265641324 T^{6} + 18564443065128 T^{8} + 18576680291578980 T^{10} + 15937813488798672705 T^{12} + \)\(12\!\cdots\!85\)\( T^{14} + \)\(81\!\cdots\!47\)\( T^{16} + \)\(50\!\cdots\!84\)\( T^{18} + \)\(27\!\cdots\!12\)\( T^{20} + \)\(50\!\cdots\!84\)\( p^{4} T^{22} + \)\(81\!\cdots\!47\)\( p^{8} T^{24} + \)\(12\!\cdots\!85\)\( p^{12} T^{26} + 15937813488798672705 p^{16} T^{28} + 18576680291578980 p^{20} T^{30} + 18564443065128 p^{24} T^{32} + 15265641324 p^{28} T^{34} + 9699319 p^{32} T^{36} + 4271 p^{36} T^{38} + p^{40} T^{40} \) | |
| 29 | \( ( 1 + 57 T + 5 p^{2} T^{2} + 177954 T^{3} + 7723410 T^{4} + 238187670 T^{5} + 8158303479 T^{6} + 199374160911 T^{7} + 6268497118587 T^{8} + 144562226975880 T^{9} + 4822948000687740 T^{10} + 144562226975880 p^{2} T^{11} + 6268497118587 p^{4} T^{12} + 199374160911 p^{6} T^{13} + 8158303479 p^{8} T^{14} + 238187670 p^{10} T^{15} + 7723410 p^{12} T^{16} + 177954 p^{14} T^{17} + 5 p^{18} T^{18} + 57 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 14 T - 2435 T^{2} + 33018 T^{3} + 2500473 T^{4} - 33845382 T^{5} - 2297474712 T^{6} + 51479351610 T^{7} + 2272325052621 T^{8} - 34459228806056 T^{9} - 1978149011082983 T^{10} - 34459228806056 p^{2} T^{11} + 2272325052621 p^{4} T^{12} + 51479351610 p^{6} T^{13} - 2297474712 p^{8} T^{14} - 33845382 p^{10} T^{15} + 2500473 p^{12} T^{16} + 33018 p^{14} T^{17} - 2435 p^{16} T^{18} - 14 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 37 | \( ( 1 + 4924 T^{2} + 10243809 T^{4} + 16042067436 T^{6} + 31004044428990 T^{8} + 52347355045062480 T^{10} + 31004044428990 p^{4} T^{12} + 16042067436 p^{8} T^{14} + 10243809 p^{12} T^{16} + 4924 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 51 T + 6218 T^{2} + 272901 T^{3} + 17317464 T^{4} + 691875705 T^{5} + 38745648894 T^{6} + 1544954154321 T^{7} + 88564431342915 T^{8} + 3421638483866670 T^{9} + 170859779134869912 T^{10} + 3421638483866670 p^{2} T^{11} + 88564431342915 p^{4} T^{12} + 1544954154321 p^{6} T^{13} + 38745648894 p^{8} T^{14} + 691875705 p^{10} T^{15} + 17317464 p^{12} T^{16} + 272901 p^{14} T^{17} + 6218 p^{16} T^{18} + 51 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 43 | \( 1 - 5998 T^{2} + 10190950 T^{4} + 6350330652 T^{6} - 16662509674053 T^{8} - 100727177271974796 T^{10} + \)\(26\!\cdots\!20\)\( T^{12} + \)\(87\!\cdots\!62\)\( p T^{14} - \)\(64\!\cdots\!07\)\( T^{16} - \)\(52\!\cdots\!40\)\( T^{18} + \)\(36\!\cdots\!10\)\( T^{20} - \)\(52\!\cdots\!40\)\( p^{4} T^{22} - \)\(64\!\cdots\!07\)\( p^{8} T^{24} + \)\(87\!\cdots\!62\)\( p^{13} T^{26} + \)\(26\!\cdots\!20\)\( p^{16} T^{28} - 100727177271974796 p^{20} T^{30} - 16662509674053 p^{24} T^{32} + 6350330652 p^{28} T^{34} + 10190950 p^{32} T^{36} - 5998 p^{36} T^{38} + p^{40} T^{40} \) | |
| 47 | \( 1 + 7817 T^{2} + 36357721 T^{4} + 148795805886 T^{6} + 543514436980284 T^{8} + 1782131739024199722 T^{10} + \)\(53\!\cdots\!31\)\( T^{12} + \)\(15\!\cdots\!83\)\( T^{14} + \)\(38\!\cdots\!47\)\( T^{16} + \)\(94\!\cdots\!52\)\( T^{18} + \)\(21\!\cdots\!32\)\( T^{20} + \)\(94\!\cdots\!52\)\( p^{4} T^{22} + \)\(38\!\cdots\!47\)\( p^{8} T^{24} + \)\(15\!\cdots\!83\)\( p^{12} T^{26} + \)\(53\!\cdots\!31\)\( p^{16} T^{28} + 1782131739024199722 p^{20} T^{30} + 543514436980284 p^{24} T^{32} + 148795805886 p^{28} T^{34} + 36357721 p^{32} T^{36} + 7817 p^{36} T^{38} + p^{40} T^{40} \) | |
| 53 | \( ( 1 - 13538 T^{2} + 94114761 T^{4} - 452725145712 T^{6} + 1696657909433478 T^{8} - 5219610938199252060 T^{10} + 1696657909433478 p^{4} T^{12} - 452725145712 p^{8} T^{14} + 94114761 p^{12} T^{16} - 13538 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 59 | \( ( 1 - 60 T + 8648 T^{2} - 446880 T^{3} + 33056133 T^{4} - 1785454104 T^{5} + 116384281698 T^{6} - 9809460008220 T^{7} + 568264098086073 T^{8} - 45786010928926296 T^{9} + 2406236323471651194 T^{10} - 45786010928926296 p^{2} T^{11} + 568264098086073 p^{4} T^{12} - 9809460008220 p^{6} T^{13} + 116384281698 p^{8} T^{14} - 1785454104 p^{10} T^{15} + 33056133 p^{12} T^{16} - 446880 p^{14} T^{17} + 8648 p^{16} T^{18} - 60 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 25 T - 14957 T^{2} - 348996 T^{3} + 126854706 T^{4} + 2452843674 T^{5} - 783135738441 T^{6} - 8955604330725 T^{7} + 3901842805497177 T^{8} + 13994293939965622 T^{9} - 15936958107064238072 T^{10} + 13994293939965622 p^{2} T^{11} + 3901842805497177 p^{4} T^{12} - 8955604330725 p^{6} T^{13} - 783135738441 p^{8} T^{14} + 2452843674 p^{10} T^{15} + 126854706 p^{12} T^{16} - 348996 p^{14} T^{17} - 14957 p^{16} T^{18} + 25 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 67 | \( 1 - 13993 T^{2} + 84984610 T^{4} - 148579225383 T^{6} - 1233944242245888 T^{8} + 8862768246217696269 T^{10} - \)\(95\!\cdots\!90\)\( T^{12} - \)\(12\!\cdots\!79\)\( T^{14} + \)\(50\!\cdots\!63\)\( T^{16} + \)\(15\!\cdots\!70\)\( T^{18} - \)\(17\!\cdots\!20\)\( T^{20} + \)\(15\!\cdots\!70\)\( p^{4} T^{22} + \)\(50\!\cdots\!63\)\( p^{8} T^{24} - \)\(12\!\cdots\!79\)\( p^{12} T^{26} - \)\(95\!\cdots\!90\)\( p^{16} T^{28} + 8862768246217696269 p^{20} T^{30} - 1233944242245888 p^{24} T^{32} - 148579225383 p^{28} T^{34} + 84984610 p^{32} T^{36} - 13993 p^{36} T^{38} + p^{40} T^{40} \) | |
| 71 | \( ( 1 - 22654 T^{2} + 323616969 T^{4} - 3114753032244 T^{6} + 23029092307426278 T^{8} - \)\(13\!\cdots\!80\)\( T^{10} + 23029092307426278 p^{4} T^{12} - 3114753032244 p^{8} T^{14} + 323616969 p^{12} T^{16} - 22654 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 24544 T^{2} + 237017538 T^{4} + 776756490186 T^{6} - 4380504629537715 T^{8} - 50162655079102511988 T^{10} - 4380504629537715 p^{4} T^{12} + 776756490186 p^{8} T^{14} + 237017538 p^{12} T^{16} + 24544 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 64 T - 13535 T^{2} + 190704 T^{3} + 135894801 T^{4} + 1688466672 T^{5} - 483692591664 T^{6} - 36687396347304 T^{7} - 406382983361871 T^{8} + 911298470640872 p T^{9} + 22029468032302837729 T^{10} + 911298470640872 p^{3} T^{11} - 406382983361871 p^{4} T^{12} - 36687396347304 p^{6} T^{13} - 483692591664 p^{8} T^{14} + 1688466672 p^{10} T^{15} + 135894801 p^{12} T^{16} + 190704 p^{14} T^{17} - 13535 p^{16} T^{18} - 64 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 83 | \( 1 + 43511 T^{2} + 925905631 T^{4} + 13815732750996 T^{6} + 171641030380442904 T^{8} + \)\(18\!\cdots\!60\)\( T^{10} + \)\(18\!\cdots\!69\)\( T^{12} + \)\(16\!\cdots\!49\)\( T^{14} + \)\(13\!\cdots\!39\)\( T^{16} + \)\(10\!\cdots\!64\)\( T^{18} + \)\(74\!\cdots\!92\)\( T^{20} + \)\(10\!\cdots\!64\)\( p^{4} T^{22} + \)\(13\!\cdots\!39\)\( p^{8} T^{24} + \)\(16\!\cdots\!49\)\( p^{12} T^{26} + \)\(18\!\cdots\!69\)\( p^{16} T^{28} + \)\(18\!\cdots\!60\)\( p^{20} T^{30} + 171641030380442904 p^{24} T^{32} + 13815732750996 p^{28} T^{34} + 925905631 p^{32} T^{36} + 43511 p^{36} T^{38} + p^{40} T^{40} \) | |
| 89 | \( ( 1 - 54865 T^{2} + 1484411112 T^{4} - 25872571967043 T^{6} + 320148637671623043 T^{8} - \)\(29\!\cdots\!16\)\( T^{10} + 320148637671623043 p^{4} T^{12} - 25872571967043 p^{8} T^{14} + 1484411112 p^{12} T^{16} - 54865 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 97 | \( 1 - 68512 T^{2} + 2461402798 T^{4} - 60984637403196 T^{6} + 1164357161640028191 T^{8} - \)\(18\!\cdots\!20\)\( T^{10} + \)\(24\!\cdots\!92\)\( T^{12} - \)\(28\!\cdots\!52\)\( T^{14} + \)\(30\!\cdots\!09\)\( T^{16} - \)\(30\!\cdots\!44\)\( T^{18} + \)\(28\!\cdots\!86\)\( T^{20} - \)\(30\!\cdots\!44\)\( p^{4} T^{22} + \)\(30\!\cdots\!09\)\( p^{8} T^{24} - \)\(28\!\cdots\!52\)\( p^{12} T^{26} + \)\(24\!\cdots\!92\)\( p^{16} T^{28} - \)\(18\!\cdots\!20\)\( p^{20} T^{30} + 1164357161640028191 p^{24} T^{32} - 60984637403196 p^{28} T^{34} + 2461402798 p^{32} T^{36} - 68512 p^{36} T^{38} + p^{40} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.00993563310571080036944171857, −1.94491205023782551853644420804, −1.93843326114062922702217919540, −1.90933799939115721989664211274, −1.85769601441341385822942628235, −1.82356801042679196007936017925, −1.68983435835903265665761853094, −1.66583228925815375764297740481, −1.65009830847523724148209050319, −1.31828515013168977544875717096, −1.23209268191118964648571645046, −1.19535453431550685129730210221, −1.18697749320751031853799859386, −1.12210860002363616666357859158, −1.04267979020904214421833778584, −0.876028799275523885567374825456, −0.820253119969082507715094432384, −0.73038759905701303450473956433, −0.68920368734194563569622791691, −0.64581412225438211822541377312, −0.63567369491801050999517334574, −0.30833106365068676613671524981, −0.11224429400179295998414035093, −0.10392727744404233115864142860, −0.07783461385387076316882197500, 0.07783461385387076316882197500, 0.10392727744404233115864142860, 0.11224429400179295998414035093, 0.30833106365068676613671524981, 0.63567369491801050999517334574, 0.64581412225438211822541377312, 0.68920368734194563569622791691, 0.73038759905701303450473956433, 0.820253119969082507715094432384, 0.876028799275523885567374825456, 1.04267979020904214421833778584, 1.12210860002363616666357859158, 1.18697749320751031853799859386, 1.19535453431550685129730210221, 1.23209268191118964648571645046, 1.31828515013168977544875717096, 1.65009830847523724148209050319, 1.66583228925815375764297740481, 1.68983435835903265665761853094, 1.82356801042679196007936017925, 1.85769601441341385822942628235, 1.90933799939115721989664211274, 1.93843326114062922702217919540, 1.94491205023782551853644420804, 2.00993563310571080036944171857
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.