Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 4-s − 4·6-s − 4·7-s − 9-s + 6·11-s − 2·12-s + 7·13-s − 8·14-s + 5·17-s − 2·18-s − 2·19-s + 8·21-s + 12·22-s − 15·23-s − 12·25-s + 14·26-s + 14·27-s − 4·28-s − 19·29-s + 13·31-s − 2·32-s − 12·33-s + 10·34-s − 36-s − 40·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s − 1.51·7-s − 1/3·9-s + 1.80·11-s − 0.577·12-s + 1.94·13-s − 2.13·14-s + 1.21·17-s − 0.471·18-s − 0.458·19-s + 1.74·21-s + 2.55·22-s − 3.12·23-s − 2.39·25-s + 2.74·26-s + 2.69·27-s − 0.755·28-s − 3.52·29-s + 2.33·31-s − 0.353·32-s − 2.08·33-s + 1.71·34-s − 1/6·36-s − 6.57·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 31^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00360869\) |
| Root analytic conductor: | \(0.703613\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4374080377\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4374080377\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 319 p T^{5} + 48 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + 2 T + 5 T^{2} - 2 T^{3} - 16 T^{4} - 44 T^{5} - 7 p T^{6} + 32 p T^{7} + 271 T^{8} + 32 p^{2} T^{9} - 7 p^{3} T^{10} - 44 p^{3} T^{11} - 16 p^{4} T^{12} - 2 p^{5} T^{13} + 5 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 6 T^{2} + 3 p T^{3} + 19 T^{4} + 3 p^{2} T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 7 | \( ( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 10 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( 1 - 6 T + 3 p T^{2} - 201 T^{3} + 937 T^{4} - 3825 T^{5} + 16437 T^{6} - 5448 p T^{7} + 18005 p T^{8} - 5448 p^{2} T^{9} + 16437 p^{2} T^{10} - 3825 p^{3} T^{11} + 937 p^{4} T^{12} - 201 p^{5} T^{13} + 3 p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 7 T + 24 T^{2} - 109 T^{3} + 496 T^{4} - 1996 T^{5} + 9030 T^{6} - 37400 T^{7} + 135607 T^{8} - 37400 p T^{9} + 9030 p^{2} T^{10} - 1996 p^{3} T^{11} + 496 p^{4} T^{12} - 109 p^{5} T^{13} + 24 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 5 T - 7 T^{2} + 30 T^{3} + 485 T^{4} - 530 T^{5} - 11652 T^{6} + 23475 T^{7} + 69679 T^{8} + 23475 p T^{9} - 11652 p^{2} T^{10} - 530 p^{3} T^{11} + 485 p^{4} T^{12} + 30 p^{5} T^{13} - 7 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 2 T + 5 T^{2} - 25 T^{3} - 105 T^{4} - 1189 T^{5} + 4287 T^{6} + 11220 T^{7} + 135775 T^{8} + 11220 p T^{9} + 4287 p^{2} T^{10} - 1189 p^{3} T^{11} - 105 p^{4} T^{12} - 25 p^{5} T^{13} + 5 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 15 T + 114 T^{2} + 495 T^{3} + 592 T^{4} - 2490 T^{5} + 13632 T^{6} + 349800 T^{7} + 2485405 T^{8} + 349800 p T^{9} + 13632 p^{2} T^{10} - 2490 p^{3} T^{11} + 592 p^{4} T^{12} + 495 p^{5} T^{13} + 114 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 19 T + 101 T^{2} - 510 T^{3} - 8767 T^{4} - 36410 T^{5} + 33084 T^{6} + 1024269 T^{7} + 6539383 T^{8} + 1024269 p T^{9} + 33084 p^{2} T^{10} - 36410 p^{3} T^{11} - 8767 p^{4} T^{12} - 510 p^{5} T^{13} + 101 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 + 20 T + 284 T^{2} + 2565 T^{3} + 18487 T^{4} + 2565 p T^{5} + 284 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 7 T - 37 T^{2} + 66 T^{3} + 4757 T^{4} + 13960 T^{5} - 40728 T^{6} + 470187 T^{7} + 7783975 T^{8} + 470187 p T^{9} - 40728 p^{2} T^{10} + 13960 p^{3} T^{11} + 4757 p^{4} T^{12} + 66 p^{5} T^{13} - 37 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 12 T + 14 T^{2} + 236 T^{3} + 991 T^{4} - 9516 T^{5} - 85540 T^{6} + 935280 T^{7} - 4999843 T^{8} + 935280 p T^{9} - 85540 p^{2} T^{10} - 9516 p^{3} T^{11} + 991 p^{4} T^{12} + 236 p^{5} T^{13} + 14 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( ( 1 + 6 T - 11 T^{2} + 222 T^{3} + 3559 T^{4} + 222 p T^{5} - 11 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 53 | \( 1 - 9 T - 96 T^{2} + 1230 T^{3} + 406 T^{4} - 70077 T^{5} + 497550 T^{6} + 1453452 T^{7} - 39269633 T^{8} + 1453452 p T^{9} + 497550 p^{2} T^{10} - 70077 p^{3} T^{11} + 406 p^{4} T^{12} + 1230 p^{5} T^{13} - 96 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 18 T + 107 T^{2} + 102 T^{3} + 102 T^{4} + 69300 T^{5} + 840883 T^{6} + 3496446 T^{7} + 10651475 T^{8} + 3496446 p T^{9} + 840883 p^{2} T^{10} + 69300 p^{3} T^{11} + 102 p^{4} T^{12} + 102 p^{5} T^{13} + 107 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( ( 1 - 6 T + 216 T^{2} - 1012 T^{3} + 19161 T^{4} - 1012 p T^{5} + 216 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 13 T + 282 T^{2} + 2445 T^{3} + 28981 T^{4} + 2445 p T^{5} + 282 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 25 T + 170 T^{2} - 1410 T^{3} - 35806 T^{4} - 314705 T^{5} - 659400 T^{6} + 21181560 T^{7} + 295190251 T^{8} + 21181560 p T^{9} - 659400 p^{2} T^{10} - 314705 p^{3} T^{11} - 35806 p^{4} T^{12} - 1410 p^{5} T^{13} + 170 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 35 T + 424 T^{2} - 1075 T^{3} - 16678 T^{4} + 61060 T^{5} + 844832 T^{6} + 592550 T^{7} - 87991445 T^{8} + 592550 p T^{9} + 844832 p^{2} T^{10} + 61060 p^{3} T^{11} - 16678 p^{4} T^{12} - 1075 p^{5} T^{13} + 424 p^{6} T^{14} - 35 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 6 T - 103 T^{2} + 683 T^{3} + 12157 T^{4} - 23205 T^{5} - 1606927 T^{6} + 723636 T^{7} + 131466275 T^{8} + 723636 p T^{9} - 1606927 p^{2} T^{10} - 23205 p^{3} T^{11} + 12157 p^{4} T^{12} + 683 p^{5} T^{13} - 103 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 12 T - 29 T^{2} + 774 T^{3} - 2741 T^{4} + 774 p T^{5} - 29 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 7 T + 200 T^{2} - 420 T^{3} + 6050 T^{4} - 161339 T^{5} + 1081842 T^{6} + 1450050 T^{7} + 229631995 T^{8} + 1450050 p T^{9} + 1081842 p^{2} T^{10} - 161339 p^{3} T^{11} + 6050 p^{4} T^{12} - 420 p^{5} T^{13} + 200 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 26 T + 216 T^{2} - 1178 T^{3} + 24961 T^{4} - 207098 T^{5} - 1010280 T^{6} + 10171100 T^{7} + 45612217 T^{8} + 10171100 p T^{9} - 1010280 p^{2} T^{10} - 207098 p^{3} T^{11} + 24961 p^{4} T^{12} - 1178 p^{5} T^{13} + 216 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−7.12059455256164925323394830305, −7.00759551180591823631475234829, −6.89723161661783113917978355563, −6.40721647464635013961024553604, −6.39252061699042314790989380678, −6.25572019484939935429637141249, −6.04590581038875975849091816613, −6.02729670419625671369950665219, −5.95473384180407717815223696556, −5.57341494907444879961472610434, −5.57054349070511346417697025475, −5.16552983230636990996384603552, −4.92051164826359304092076805227, −4.91977911745576872572309769370, −4.59140907646697592628821373139, −4.06637469604218327145211726045, −4.06236322350928226466018868623, −3.71745698967032881466807175778, −3.70919692856038466472340497286, −3.42614523253195833193741103486, −3.31361775191376484645838891897, −3.25657413500718852002690975402, −2.15201030841526748047471868594, −1.97597932975552870334453809057, −1.74554332547384636926130650134, 1.74554332547384636926130650134, 1.97597932975552870334453809057, 2.15201030841526748047471868594, 3.25657413500718852002690975402, 3.31361775191376484645838891897, 3.42614523253195833193741103486, 3.70919692856038466472340497286, 3.71745698967032881466807175778, 4.06236322350928226466018868623, 4.06637469604218327145211726045, 4.59140907646697592628821373139, 4.91977911745576872572309769370, 4.92051164826359304092076805227, 5.16552983230636990996384603552, 5.57054349070511346417697025475, 5.57341494907444879961472610434, 5.95473384180407717815223696556, 6.02729670419625671369950665219, 6.04590581038875975849091816613, 6.25572019484939935429637141249, 6.39252061699042314790989380678, 6.40721647464635013961024553604, 6.89723161661783113917978355563, 7.00759551180591823631475234829, 7.12059455256164925323394830305
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.