Dirichlet series
| L(s) = 1 | − 2·3-s + 2·4-s + 2·5-s − 30·7-s + 9·9-s − 4·11-s − 4·12-s + 30·13-s − 4·15-s + 30·17-s − 30·19-s + 4·20-s + 60·21-s − 104·23-s + 21·25-s − 48·27-s − 60·28-s − 10·29-s + 46·31-s + 8·33-s − 60·35-s + 18·36-s + 6·37-s − 60·39-s + 250·41-s − 8·44-s + 18·45-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 1/2·4-s + 2/5·5-s − 4.28·7-s + 9-s − 0.363·11-s − 1/3·12-s + 2.30·13-s − 0.266·15-s + 1.76·17-s − 1.57·19-s + 1/5·20-s + 20/7·21-s − 4.52·23-s + 0.839·25-s − 1.77·27-s − 2.14·28-s − 0.344·29-s + 1.48·31-s + 8/33·33-s − 1.71·35-s + 1/2·36-s + 6/37·37-s − 1.53·39-s + 6.09·41-s − 0.181·44-s + 2/5·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0166748\) |
| Root analytic conductor: | \(0.774245\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.3105019971\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3105019971\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | \( 1 + 4 T - 4 p^{2} T^{3} - 6 p^{3} T^{4} - 4 p^{4} T^{5} + 4 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 3 | \( 1 + 2 T - 5 T^{2} + 20 T^{3} + 5 p^{2} T^{4} - 284 T^{5} + 31 p^{3} T^{6} + 3350 T^{7} - 3020 T^{8} + 3350 p^{2} T^{9} + 31 p^{7} T^{10} - 284 p^{6} T^{11} + 5 p^{10} T^{12} + 20 p^{10} T^{13} - 5 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} \) |
| 5 | \( 1 - 2 T - 17 T^{2} + 156 T^{3} - 77 p T^{4} - 7664 T^{5} + 29213 T^{6} + 102118 T^{7} - 966104 T^{8} + 102118 p^{2} T^{9} + 29213 p^{4} T^{10} - 7664 p^{6} T^{11} - 77 p^{9} T^{12} + 156 p^{10} T^{13} - 17 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 + 30 T + 571 T^{2} + 1180 p T^{3} + 97845 T^{4} + 141940 p T^{5} + 8888309 T^{6} + 10209790 p T^{7} + 524056964 T^{8} + 10209790 p^{3} T^{9} + 8888309 p^{4} T^{10} + 141940 p^{7} T^{11} + 97845 p^{8} T^{12} + 1180 p^{11} T^{13} + 571 p^{12} T^{14} + 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 30 T + 651 T^{2} - 6800 T^{3} + 48375 T^{4} + 238980 T^{5} - 3389311 T^{6} + 64452890 T^{7} - 51252296 T^{8} + 64452890 p^{2} T^{9} - 3389311 p^{4} T^{10} + 238980 p^{6} T^{11} + 48375 p^{8} T^{12} - 6800 p^{10} T^{13} + 651 p^{12} T^{14} - 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 30 T + 675 T^{2} - 9160 T^{3} + 193599 T^{4} - 3443220 T^{5} + 74881625 T^{6} - 864455750 T^{7} + 15090566536 T^{8} - 864455750 p^{2} T^{9} + 74881625 p^{4} T^{10} - 3443220 p^{6} T^{11} + 193599 p^{8} T^{12} - 9160 p^{10} T^{13} + 675 p^{12} T^{14} - 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 30 T + 999 T^{2} + 20440 T^{3} + 415005 T^{4} + 6646680 T^{5} + 145579961 T^{6} + 3061994690 T^{7} + 58504536724 T^{8} + 3061994690 p^{2} T^{9} + 145579961 p^{4} T^{10} + 6646680 p^{6} T^{11} + 415005 p^{8} T^{12} + 20440 p^{10} T^{13} + 999 p^{12} T^{14} + 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( ( 1 + 52 T + 2240 T^{2} + 63372 T^{3} + 1720014 T^{4} + 63372 p^{2} T^{5} + 2240 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 29 | \( 1 + 10 T + 1515 T^{2} - 2160 p T^{3} - 142561 T^{4} - 111518460 T^{5} + 1681998945 T^{6} - 18301053550 T^{7} + 3803814204136 T^{8} - 18301053550 p^{2} T^{9} + 1681998945 p^{4} T^{10} - 111518460 p^{6} T^{11} - 142561 p^{8} T^{12} - 2160 p^{11} T^{13} + 1515 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 - 46 T - 815 T^{2} + 85900 T^{3} - 728375 T^{4} + 2317612 T^{5} - 1257533057 T^{6} - 34712738410 T^{7} + 3877854489500 T^{8} - 34712738410 p^{2} T^{9} - 1257533057 p^{4} T^{10} + 2317612 p^{6} T^{11} - 728375 p^{8} T^{12} + 85900 p^{10} T^{13} - 815 p^{12} T^{14} - 46 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 6 T - 2421 T^{2} - 14136 T^{3} + 5054271 T^{4} - 6253908 T^{5} - 8619590399 T^{6} + 1373683362 T^{7} + 13772803179432 T^{8} + 1373683362 p^{2} T^{9} - 8619590399 p^{4} T^{10} - 6253908 p^{6} T^{11} + 5054271 p^{8} T^{12} - 14136 p^{10} T^{13} - 2421 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 250 T + 27975 T^{2} - 1814420 T^{3} + 77101799 T^{4} - 2779318000 T^{5} + 139016689365 T^{6} - 207457086610 p T^{7} + 246006397816 p^{2} T^{8} - 207457086610 p^{3} T^{9} + 139016689365 p^{4} T^{10} - 2779318000 p^{6} T^{11} + 77101799 p^{8} T^{12} - 1814420 p^{10} T^{13} + 27975 p^{12} T^{14} - 250 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 - 11160 T^{2} + 57008924 T^{4} - 179309232040 T^{6} + 390998083541766 T^{8} - 179309232040 p^{4} T^{10} + 57008924 p^{8} T^{12} - 11160 p^{12} T^{14} + p^{16} T^{16} \) | |
| 47 | \( 1 + 54 T - 1941 T^{2} - 123836 T^{3} + 6739581 T^{4} + 148592292 T^{5} - 27255791899 T^{6} - 305251818238 T^{7} + 49111755331252 T^{8} - 305251818238 p^{2} T^{9} - 27255791899 p^{4} T^{10} + 148592292 p^{6} T^{11} + 6739581 p^{8} T^{12} - 123836 p^{10} T^{13} - 1941 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 274 T + 38659 T^{2} + 3972344 T^{3} + 334778231 T^{4} + 23878670812 T^{5} + 1495056085961 T^{6} + 86289965064362 T^{7} + 4702527071102392 T^{8} + 86289965064362 p^{2} T^{9} + 1495056085961 p^{4} T^{10} + 23878670812 p^{6} T^{11} + 334778231 p^{8} T^{12} + 3972344 p^{10} T^{13} + 38659 p^{12} T^{14} + 274 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 50 T + 1263 T^{2} + 70200 T^{3} - 2470907 T^{4} + 1076735200 T^{5} - 12586111599 T^{6} + 433998162250 T^{7} + 92677785681540 T^{8} + 433998162250 p^{2} T^{9} - 12586111599 p^{4} T^{10} + 1076735200 p^{6} T^{11} - 2470907 p^{8} T^{12} + 70200 p^{10} T^{13} + 1263 p^{12} T^{14} - 50 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 50 T + 10707 T^{2} - 429000 T^{3} + 36206983 T^{4} - 875034700 T^{5} - 30917305831 T^{6} + 2941568049750 T^{7} - 516150507559720 T^{8} + 2941568049750 p^{2} T^{9} - 30917305831 p^{4} T^{10} - 875034700 p^{6} T^{11} + 36206983 p^{8} T^{12} - 429000 p^{10} T^{13} + 10707 p^{12} T^{14} - 50 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( ( 1 - 56 T + 17612 T^{2} - 732008 T^{3} + 117930870 T^{4} - 732008 p^{2} T^{5} + 17612 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 71 | \( 1 - 54 T - 9075 T^{2} + 1051560 T^{3} - 949335 T^{4} - 6026869752 T^{5} + 391790178883 T^{6} + 11973624475590 T^{7} - 2665107650348580 T^{8} + 11973624475590 p^{2} T^{9} + 391790178883 p^{4} T^{10} - 6026869752 p^{6} T^{11} - 949335 p^{8} T^{12} + 1051560 p^{10} T^{13} - 9075 p^{12} T^{14} - 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 + 70 T + 14643 T^{2} + 877280 T^{3} + 1189311 p T^{4} + 4301905820 T^{5} + 83475543561 T^{6} + 7156391135710 T^{7} - 896454269626600 T^{8} + 7156391135710 p^{2} T^{9} + 83475543561 p^{4} T^{10} + 4301905820 p^{6} T^{11} + 1189311 p^{9} T^{12} + 877280 p^{10} T^{13} + 14643 p^{12} T^{14} + 70 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 370 T + 69785 T^{2} - 8224700 T^{3} + 599308969 T^{4} - 15803939780 T^{5} - 2371050090825 T^{6} + 423925649678930 T^{7} - 40585609775890084 T^{8} + 423925649678930 p^{2} T^{9} - 2371050090825 p^{4} T^{10} - 15803939780 p^{6} T^{11} + 599308969 p^{8} T^{12} - 8224700 p^{10} T^{13} + 69785 p^{12} T^{14} - 370 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 + 150 T + 16661 T^{2} + 847320 T^{3} + 56225865 T^{4} + 2891415120 T^{5} + 366983457739 T^{6} - 42655302510 T^{7} + 368491437191804 T^{8} - 42655302510 p^{2} T^{9} + 366983457739 p^{4} T^{10} + 2891415120 p^{6} T^{11} + 56225865 p^{8} T^{12} + 847320 p^{10} T^{13} + 16661 p^{12} T^{14} + 150 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( ( 1 - 12 T + 11848 T^{2} + 1140876 T^{3} + 39403230 T^{4} + 1140876 p^{2} T^{5} + 11848 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 97 | \( 1 + 18 T - 17345 T^{2} - 545820 T^{3} + 224453335 T^{4} + 2169903904 T^{5} - 2647455520643 T^{6} - 4734820613590 T^{7} + 27980585557864280 T^{8} - 4734820613590 p^{2} T^{9} - 2647455520643 p^{4} T^{10} + 2169903904 p^{6} T^{11} + 224453335 p^{8} T^{12} - 545820 p^{10} T^{13} - 17345 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−9.006854138060750659350263416905, −8.985967301909044977999039745080, −8.401795627393784154455386662609, −8.311746418073968701155347532633, −8.009644472684696138628166266376, −7.910674900902620417643238420970, −7.78959868017045901483258464012, −7.39125816057870753208758534560, −7.22680654322096989191572480543, −6.71124719653999443395347743901, −6.37008209674924873466527623609, −6.35569735595965537330627012631, −6.21795301850528591583957092807, −6.21278158787176772724109044382, −6.11735529049354510452008930543, −5.66885838076168589933895992365, −5.61968806668419373292087975594, −4.83702689320240526826313255394, −4.43574587513077144382252248678, −4.18630778632312369780562693955, −3.65684294665794379353747902414, −3.52873176118917844243502079164, −3.47050307664269194133164855593, −2.72746729182409857954834742013, −2.16544580626431145453125438091, 2.16544580626431145453125438091, 2.72746729182409857954834742013, 3.47050307664269194133164855593, 3.52873176118917844243502079164, 3.65684294665794379353747902414, 4.18630778632312369780562693955, 4.43574587513077144382252248678, 4.83702689320240526826313255394, 5.61968806668419373292087975594, 5.66885838076168589933895992365, 6.11735529049354510452008930543, 6.21278158787176772724109044382, 6.21795301850528591583957092807, 6.35569735595965537330627012631, 6.37008209674924873466527623609, 6.71124719653999443395347743901, 7.22680654322096989191572480543, 7.39125816057870753208758534560, 7.78959868017045901483258464012, 7.910674900902620417643238420970, 8.009644472684696138628166266376, 8.311746418073968701155347532633, 8.401795627393784154455386662609, 8.985967301909044977999039745080, 9.006854138060750659350263416905
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.