Dirichlet series
| L(s) = 1 | − 2·2-s + 12·3-s + 7·4-s − 24·6-s + 16·7-s − 6·8-s + 78·9-s + 20·11-s + 84·12-s − 32·14-s + 18·16-s + 18·17-s − 156·18-s + 192·21-s − 40·22-s − 62·23-s − 72·24-s + 360·27-s + 112·28-s − 100·29-s − 126·31-s + 240·33-s − 36·34-s + 546·36-s + 80·37-s − 384·42-s − 352·43-s + ⋯ |
| L(s) = 1 | − 2-s + 4·3-s + 7/4·4-s − 4·6-s + 16/7·7-s − 3/4·8-s + 26/3·9-s + 1.81·11-s + 7·12-s − 2.28·14-s + 9/8·16-s + 1.05·17-s − 8.66·18-s + 64/7·21-s − 1.81·22-s − 2.69·23-s − 3·24-s + 40/3·27-s + 4·28-s − 3.44·29-s − 4.06·31-s + 7.27·33-s − 1.05·34-s + 91/6·36-s + 2.16·37-s − 9.14·42-s − 8.18·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.75369\times 10^{9}\) |
| Root analytic conductor: | \(3.78222\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(17.22552629\) |
| \(L(\frac12)\) | \(\approx\) | \(17.22552629\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 - 16 T + 109 T^{2} - 16 p^{2} T^{3} + 136 p^{2} T^{4} - 16 p^{4} T^{5} + 109 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 2 | \( 1 + p T - 3 T^{2} - 7 p T^{3} - 13 T^{4} + 9 p T^{5} + 35 p T^{6} + 11 p^{2} T^{7} - 39 p^{2} T^{8} + 11 p^{4} T^{9} + 35 p^{5} T^{10} + 9 p^{7} T^{11} - 13 p^{8} T^{12} - 7 p^{11} T^{13} - 3 p^{12} T^{14} + p^{15} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 8003440 p^{2} T^{9} - 8882912 p^{4} T^{10} - 519480 p^{6} T^{11} + 57131 p^{8} T^{12} + 2960 p^{10} T^{13} - 147 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 8798048 p^{4} T^{10} + 39826 p^{8} T^{12} - 188 p^{12} T^{14} + p^{16} T^{16} \) | |
| 17 | \( 1 - 18 T + 766 T^{2} - 11844 T^{3} + 262438 T^{4} - 1906254 T^{5} + 22625848 T^{6} + 382636314 T^{7} - 5169040877 T^{8} + 382636314 p^{2} T^{9} + 22625848 p^{4} T^{10} - 1906254 p^{6} T^{11} + 262438 p^{8} T^{12} - 11844 p^{10} T^{13} + 766 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 598 T^{2} + 183481 T^{4} - 682560 T^{5} - 48973562 T^{6} - 501474240 T^{7} - 32269961996 T^{8} - 501474240 p^{2} T^{9} - 48973562 p^{4} T^{10} - 682560 p^{6} T^{11} + 183481 p^{8} T^{12} + 598 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 + 62 T + 1497 T^{2} - 6014 T^{3} - 1196893 T^{4} - 31086552 T^{5} - 143771420 T^{6} + 13896561704 T^{7} + 513552019554 T^{8} + 13896561704 p^{2} T^{9} - 143771420 p^{4} T^{10} - 31086552 p^{6} T^{11} - 1196893 p^{8} T^{12} - 6014 p^{10} T^{13} + 1497 p^{12} T^{14} + 62 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( ( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 15850 p^{2} T^{5} + 1234 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 31 | \( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1243487527488 p^{2} T^{9} + 38160094402 p^{4} T^{10} + 1079090100 p^{6} T^{11} + 27206317 p^{8} T^{12} + 578466 p^{10} T^{13} + 9883 p^{12} T^{14} + 126 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 80 T + 1194 T^{2} - 28960 T^{3} + 3461705 T^{4} + 28416960 T^{5} - 6540374054 T^{6} + 198858748720 T^{7} - 6603314864556 T^{8} + 198858748720 p^{2} T^{9} - 6540374054 p^{4} T^{10} + 28416960 p^{6} T^{11} + 3461705 p^{8} T^{12} - 28960 p^{10} T^{13} + 1194 p^{12} T^{14} - 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 146585251874 p^{4} T^{10} + 48877645 p^{8} T^{12} - 10106 p^{12} T^{14} + p^{16} T^{16} \) | |
| 43 | \( ( 1 + 176 T + 17017 T^{2} + 1139948 T^{3} + 56853640 T^{4} + 1139948 p^{2} T^{5} + 17017 p^{4} T^{6} + 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 47 | \( 1 - 72 T + 233 p T^{2} - 664056 T^{3} + 65519473 T^{4} - 3342900456 T^{5} + 246192812578 T^{6} - 10750018584384 T^{7} + 651041931981118 T^{8} - 10750018584384 p^{2} T^{9} + 246192812578 p^{4} T^{10} - 3342900456 p^{6} T^{11} + 65519473 p^{8} T^{12} - 664056 p^{10} T^{13} + 233 p^{13} T^{14} - 72 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 76 T - 3069 T^{2} + 443764 T^{3} + 2229785 T^{4} - 1396117872 T^{5} + 34651196266 T^{6} + 1991894657480 T^{7} - 169369280357850 T^{8} + 1991894657480 p^{2} T^{9} + 34651196266 p^{4} T^{10} - 1396117872 p^{6} T^{11} + 2229785 p^{8} T^{12} + 443764 p^{10} T^{13} - 3069 p^{12} T^{14} - 76 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 + 54 T + 122 p T^{2} + 336204 T^{3} + 19932742 T^{4} - 202333950 T^{5} - 35478676088 T^{6} - 7641841019598 T^{7} - 385856896323245 T^{8} - 7641841019598 p^{2} T^{9} - 35478676088 p^{4} T^{10} - 202333950 p^{6} T^{11} + 19932742 p^{8} T^{12} + 336204 p^{10} T^{13} + 122 p^{13} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 825650586150588 p^{2} T^{9} + 11318984386192 p^{4} T^{10} + 136250283708 p^{6} T^{11} + 1413738778 p^{8} T^{12} + 12233232 p^{10} T^{13} + 83164 p^{12} T^{14} + 396 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 184 T + 93 p T^{2} - 140176 T^{3} + 74370665 T^{4} + 7237038408 T^{5} + 24063966106 T^{6} + 15184872524680 T^{7} + 3087140085953070 T^{8} + 15184872524680 p^{2} T^{9} + 24063966106 p^{4} T^{10} + 7237038408 p^{6} T^{11} + 74370665 p^{8} T^{12} - 140176 p^{10} T^{13} + 93 p^{13} T^{14} + 184 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 846262 p^{2} T^{5} + 12166 p^{4} T^{6} - 82 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 + 348 T + 68263 T^{2} + 9707460 T^{3} + 1072498525 T^{4} + 96253557984 T^{5} + 7434307414846 T^{6} + 526544361727584 T^{7} + 37365046682274814 T^{8} + 526544361727584 p^{2} T^{9} + 7434307414846 p^{4} T^{10} + 96253557984 p^{6} T^{11} + 1072498525 p^{8} T^{12} + 9707460 p^{10} T^{13} + 68263 p^{12} T^{14} + 348 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 + 206 T + 5583 T^{2} - 659438 T^{3} + 124066817 T^{4} + 19494076044 T^{5} + 428008398310 T^{6} + 33090623674568 T^{7} + 7605703397631354 T^{8} + 33090623674568 p^{2} T^{9} + 428008398310 p^{4} T^{10} + 19494076044 p^{6} T^{11} + 124066817 p^{8} T^{12} - 659438 p^{10} T^{13} + 5583 p^{12} T^{14} + 206 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} - 2182268545136 p^{4} T^{10} + 223804480 p^{8} T^{12} - 20672 p^{12} T^{14} + p^{16} T^{16} \) | |
| 89 | \( 1 - 282 T + 59686 T^{2} - 9356196 T^{3} + 1240796086 T^{4} - 138656838366 T^{5} + 14271061565800 T^{6} - 1337157406377822 T^{7} + 121622616146107507 T^{8} - 1337157406377822 p^{2} T^{9} + 14271061565800 p^{4} T^{10} - 138656838366 p^{6} T^{11} + 1240796086 p^{8} T^{12} - 9356196 p^{10} T^{13} + 59686 p^{12} T^{14} - 282 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} - 12414040936928 p^{4} T^{10} + 925514428 p^{8} T^{12} - 44576 p^{12} T^{14} + p^{16} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.45402874659706640609982343020, −4.31679819756314617416688229937, −4.24239326376909010433885054288, −3.93632681219449543128213886483, −3.88175011149880718508057262770, −3.60261001931071849129954167560, −3.53230230708504958243864191889, −3.44325939486487390888568724395, −3.31370973235140614577313672006, −3.24342761144266413238268276614, −3.13589152366151533150933728558, −2.83514748170654609623080654833, −2.64966986921072864174717927180, −2.49123339022612096553975282553, −2.20547000091597056011422785065, −2.05443930185927962462508569275, −1.81515046365651662163107298967, −1.74037098965819372079639817234, −1.70438845572569199623789452088, −1.59914813550082804767229933580, −1.47447484808991717827002218735, −1.37968764750634705699301343898, −1.28739852689668380067175168414, −0.35562841508001243629573854875, −0.21349187403448898163993581246, 0.21349187403448898163993581246, 0.35562841508001243629573854875, 1.28739852689668380067175168414, 1.37968764750634705699301343898, 1.47447484808991717827002218735, 1.59914813550082804767229933580, 1.70438845572569199623789452088, 1.74037098965819372079639817234, 1.81515046365651662163107298967, 2.05443930185927962462508569275, 2.20547000091597056011422785065, 2.49123339022612096553975282553, 2.64966986921072864174717927180, 2.83514748170654609623080654833, 3.13589152366151533150933728558, 3.24342761144266413238268276614, 3.31370973235140614577313672006, 3.44325939486487390888568724395, 3.53230230708504958243864191889, 3.60261001931071849129954167560, 3.88175011149880718508057262770, 3.93632681219449543128213886483, 4.24239326376909010433885054288, 4.31679819756314617416688229937, 4.45402874659706640609982343020
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.