Dirichlet series
| L(s) = 1 | + 112·3-s + 887·4-s − 7.15e3·7-s + 8.23e3·9-s + 9.93e4·12-s + 5.54e4·13-s + 4.13e5·16-s − 2.31e5·19-s − 8.01e5·21-s + 5.65e5·27-s − 6.34e6·28-s + 8.81e5·31-s + 7.30e6·36-s − 4.67e6·37-s + 6.21e6·39-s − 7.73e6·43-s + 4.62e7·48-s + 1.45e6·49-s + 4.91e7·52-s − 2.59e7·57-s + 2.24e7·61-s − 5.89e7·63-s + 1.42e8·64-s + 4.66e7·67-s + 1.29e8·73-s − 2.05e8·76-s + 1.62e8·79-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 3.46·4-s − 2.98·7-s + 1.25·9-s + 4.79·12-s + 1.94·13-s + 6.30·16-s − 1.77·19-s − 4.12·21-s + 1.06·27-s − 10.3·28-s + 0.954·31-s + 4.34·36-s − 2.49·37-s + 2.68·39-s − 2.26·43-s + 8.71·48-s + 0.252·49-s + 6.72·52-s − 2.45·57-s + 1.61·61-s − 3.73·63-s + 8.46·64-s + 2.31·67-s + 4.57·73-s − 6.15·76-s + 4.16·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+4)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.08916\times 10^{14}\) |
| Root analytic conductor: | \(5.52751\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} ,\ ( \ : [4]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(1.346824596\) |
| \(L(\frac12)\) | \(\approx\) | \(1.346824596\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 112 T + 479 p^{2} T^{2} - 520 p^{5} T^{3} + 44032 p^{6} T^{4} - 763856 p^{8} T^{5} + 44032 p^{14} T^{6} - 520 p^{21} T^{7} + 479 p^{26} T^{8} - 112 p^{32} T^{9} + p^{40} T^{10} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - 887 T^{2} + 46699 p^{3} T^{4} - 26723843 p^{2} T^{6} + 460091803 p^{6} T^{8} - 31078774227 p^{8} T^{10} + 460091803 p^{22} T^{12} - 26723843 p^{34} T^{14} + 46699 p^{51} T^{16} - 887 p^{64} T^{18} + p^{80} T^{20} \) |
| 7 | \( ( 1 + 3578 T + 2639213 p T^{2} + 127083140 p^{3} T^{3} + 63975164068 p^{4} T^{4} + 118994426049684 p^{4} T^{5} + 63975164068 p^{12} T^{6} + 127083140 p^{19} T^{7} + 2639213 p^{25} T^{8} + 3578 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 11 | \( 1 - 1554692510 T^{2} + 1180198572184778045 T^{4} - \)\(57\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!10\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!10\)\( p^{16} T^{12} - \)\(57\!\cdots\!20\)\( p^{32} T^{14} + 1180198572184778045 p^{48} T^{16} - 1554692510 p^{64} T^{18} + p^{80} T^{20} \) | |
| 13 | \( ( 1 - 27732 T + 2149414441 T^{2} - 47461070071760 T^{3} + 2040922583217943318 T^{4} - \)\(43\!\cdots\!96\)\( T^{5} + 2040922583217943318 p^{8} T^{6} - 47461070071760 p^{16} T^{7} + 2149414441 p^{24} T^{8} - 27732 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 17 | \( 1 - 26519334182 T^{2} + \)\(45\!\cdots\!57\)\( T^{4} - \)\(56\!\cdots\!32\)\( T^{6} + \)\(55\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!08\)\( p^{2} T^{10} + \)\(55\!\cdots\!22\)\( p^{16} T^{12} - \)\(56\!\cdots\!32\)\( p^{32} T^{14} + \)\(45\!\cdots\!57\)\( p^{48} T^{16} - 26519334182 p^{64} T^{18} + p^{80} T^{20} \) | |
| 19 | \( ( 1 + 115758 T + 50262268777 T^{2} + 269114327599088 T^{3} + \)\(60\!\cdots\!62\)\( T^{4} - \)\(69\!\cdots\!12\)\( T^{5} + \)\(60\!\cdots\!62\)\( p^{8} T^{6} + 269114327599088 p^{16} T^{7} + 50262268777 p^{24} T^{8} + 115758 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 23 | \( 1 - 343101374962 T^{2} + \)\(64\!\cdots\!17\)\( T^{4} - \)\(39\!\cdots\!64\)\( p T^{6} + \)\(96\!\cdots\!42\)\( T^{8} - \)\(83\!\cdots\!12\)\( T^{10} + \)\(96\!\cdots\!42\)\( p^{16} T^{12} - \)\(39\!\cdots\!64\)\( p^{33} T^{14} + \)\(64\!\cdots\!17\)\( p^{48} T^{16} - 343101374962 p^{64} T^{18} + p^{80} T^{20} \) | |
| 29 | \( 1 - 3481766656910 T^{2} + \)\(55\!\cdots\!45\)\( T^{4} - \)\(54\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!10\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{10} + \)\(37\!\cdots\!10\)\( p^{16} T^{12} - \)\(54\!\cdots\!20\)\( p^{32} T^{14} + \)\(55\!\cdots\!45\)\( p^{48} T^{16} - 3481766656910 p^{64} T^{18} + p^{80} T^{20} \) | |
| 31 | \( ( 1 - 440810 T + 2432711111345 T^{2} - 1344048327358500320 T^{3} + \)\(32\!\cdots\!10\)\( T^{4} - \)\(48\!\cdots\!92\)\( p T^{5} + \)\(32\!\cdots\!10\)\( p^{8} T^{6} - 1344048327358500320 p^{16} T^{7} + 2432711111345 p^{24} T^{8} - 440810 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 37 | \( ( 1 + 2336308 T + 7817093213961 T^{2} - 2897062774145814960 T^{3} - \)\(14\!\cdots\!02\)\( T^{4} - \)\(88\!\cdots\!76\)\( T^{5} - \)\(14\!\cdots\!02\)\( p^{8} T^{6} - 2897062774145814960 p^{16} T^{7} + 7817093213961 p^{24} T^{8} + 2336308 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 41 | \( 1 - 16585955919410 T^{2} + \)\(21\!\cdots\!45\)\( T^{4} - \)\(21\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!10\)\( p T^{8} - \)\(18\!\cdots\!52\)\( T^{10} + \)\(54\!\cdots\!10\)\( p^{17} T^{12} - \)\(21\!\cdots\!20\)\( p^{32} T^{14} + \)\(21\!\cdots\!45\)\( p^{48} T^{16} - 16585955919410 p^{64} T^{18} + p^{80} T^{20} \) | |
| 43 | \( ( 1 + 3865668 T + 32091559805911 T^{2} + 3253843559265498160 p T^{3} + \)\(70\!\cdots\!28\)\( T^{4} + \)\(19\!\cdots\!24\)\( T^{5} + \)\(70\!\cdots\!28\)\( p^{8} T^{6} + 3253843559265498160 p^{17} T^{7} + 32091559805911 p^{24} T^{8} + 3865668 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 47 | \( 1 - 32628596189522 T^{2} + \)\(89\!\cdots\!37\)\( T^{4} - \)\(34\!\cdots\!52\)\( T^{6} + \)\(88\!\cdots\!82\)\( T^{8} - \)\(20\!\cdots\!12\)\( T^{10} + \)\(88\!\cdots\!82\)\( p^{16} T^{12} - \)\(34\!\cdots\!52\)\( p^{32} T^{14} + \)\(89\!\cdots\!37\)\( p^{48} T^{16} - 32628596189522 p^{64} T^{18} + p^{80} T^{20} \) | |
| 53 | \( 1 - 109377554014022 T^{2} + \)\(15\!\cdots\!37\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!82\)\( T^{8} - \)\(60\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!82\)\( p^{16} T^{12} - \)\(11\!\cdots\!52\)\( p^{32} T^{14} + \)\(15\!\cdots\!37\)\( p^{48} T^{16} - 109377554014022 p^{64} T^{18} + p^{80} T^{20} \) | |
| 59 | \( 1 - 808545597012110 T^{2} + \)\(30\!\cdots\!45\)\( T^{4} - \)\(73\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{16} T^{12} - \)\(73\!\cdots\!20\)\( p^{32} T^{14} + \)\(30\!\cdots\!45\)\( p^{48} T^{16} - 808545597012110 p^{64} T^{18} + p^{80} T^{20} \) | |
| 61 | \( ( 1 - 11208510 T + 710787342981745 T^{2} - \)\(74\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} - \)\(20\!\cdots\!52\)\( T^{5} + \)\(22\!\cdots\!10\)\( p^{8} T^{6} - \)\(74\!\cdots\!20\)\( p^{16} T^{7} + 710787342981745 p^{24} T^{8} - 11208510 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 67 | \( ( 1 - 23323012 T + 1269159026822231 T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!84\)\( p T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!84\)\( p^{9} T^{6} - \)\(31\!\cdots\!40\)\( p^{16} T^{7} + 1269159026822231 p^{24} T^{8} - 23323012 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 71 | \( 1 - 4413680763422810 T^{2} + \)\(92\!\cdots\!45\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(89\!\cdots\!52\)\( T^{10} + \)\(12\!\cdots\!10\)\( p^{16} T^{12} - \)\(12\!\cdots\!20\)\( p^{32} T^{14} + \)\(92\!\cdots\!45\)\( p^{48} T^{16} - 4413680763422810 p^{64} T^{18} + p^{80} T^{20} \) | |
| 73 | \( ( 1 - 64982442 T + 5146332069102301 T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(90\!\cdots\!98\)\( T^{4} - \)\(24\!\cdots\!16\)\( T^{5} + \)\(90\!\cdots\!98\)\( p^{8} T^{6} - \)\(20\!\cdots\!20\)\( p^{16} T^{7} + 5146332069102301 p^{24} T^{8} - 64982442 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 79 | \( ( 1 - 81155462 T + 7827588512988617 T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(23\!\cdots\!42\)\( T^{4} - \)\(89\!\cdots\!12\)\( T^{5} + \)\(23\!\cdots\!42\)\( p^{8} T^{6} - \)\(41\!\cdots\!72\)\( p^{16} T^{7} + 7827588512988617 p^{24} T^{8} - 81155462 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 83 | \( 1 - 17295349777831282 T^{2} + \)\(14\!\cdots\!57\)\( T^{4} - \)\(72\!\cdots\!32\)\( T^{6} + \)\(25\!\cdots\!22\)\( T^{8} - \)\(67\!\cdots\!12\)\( T^{10} + \)\(25\!\cdots\!22\)\( p^{16} T^{12} - \)\(72\!\cdots\!32\)\( p^{32} T^{14} + \)\(14\!\cdots\!57\)\( p^{48} T^{16} - 17295349777831282 p^{64} T^{18} + p^{80} T^{20} \) | |
| 89 | \( 1 - 28714588631762410 T^{2} + \)\(40\!\cdots\!45\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(22\!\cdots\!10\)\( p^{16} T^{12} - \)\(36\!\cdots\!20\)\( p^{32} T^{14} + \)\(40\!\cdots\!45\)\( p^{48} T^{16} - 28714588631762410 p^{64} T^{18} + p^{80} T^{20} \) | |
| 97 | \( ( 1 - 129412862 T + 26509437329872461 T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} - \)\(24\!\cdots\!76\)\( T^{5} + \)\(32\!\cdots\!38\)\( p^{8} T^{6} - \)\(24\!\cdots\!40\)\( p^{16} T^{7} + 26509437329872461 p^{24} T^{8} - 129412862 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.98822616583802655798215740584, −3.77821792476279807980570884279, −3.73196478178487610543072330031, −3.39099327224645045318490455421, −3.28565540487221636040669744412, −3.27364243305551882032655978928, −3.27021845729774282911126782742, −3.23753120771725546213325174085, −3.08331635674378955242423175024, −2.89016179934099031533964660173, −2.35931070172291969755720450661, −2.25589152130606227332873568252, −2.20133486309970183598442499603, −2.10266691145021184419945381529, −1.98538520725752098043029734069, −1.94553829620516722549930024986, −1.78781400508475486242211900301, −1.66060702147421025963325544051, −1.06564925476390531264647565200, −1.01768345865244676290057678225, −0.823584087068358705589293501682, −0.76552306438986540600854172947, −0.75602380708347498029748471825, −0.19218995949751685574584437598, −0.04522573540101939348734107839, 0.04522573540101939348734107839, 0.19218995949751685574584437598, 0.75602380708347498029748471825, 0.76552306438986540600854172947, 0.823584087068358705589293501682, 1.01768345865244676290057678225, 1.06564925476390531264647565200, 1.66060702147421025963325544051, 1.78781400508475486242211900301, 1.94553829620516722549930024986, 1.98538520725752098043029734069, 2.10266691145021184419945381529, 2.20133486309970183598442499603, 2.25589152130606227332873568252, 2.35931070172291969755720450661, 2.89016179934099031533964660173, 3.08331635674378955242423175024, 3.23753120771725546213325174085, 3.27021845729774282911126782742, 3.27364243305551882032655978928, 3.28565540487221636040669744412, 3.39099327224645045318490455421, 3.73196478178487610543072330031, 3.77821792476279807980570884279, 3.98822616583802655798215740584
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.