Dirichlet series
| L(s) = 1 | + 3·3-s + 3·5-s − 3·7-s + 3·9-s − 18·11-s + 5·13-s + 9·15-s + 6·17-s − 9·21-s + 81·23-s + 43·25-s − 36·27-s + 69·29-s − 45·31-s − 54·33-s − 9·35-s − 20·37-s + 15·39-s + 54·41-s + 9·45-s − 207·47-s − 73·49-s + 18·51-s − 252·53-s − 54·55-s + 306·59-s + 7·61-s + ⋯ |
| L(s) = 1 | + 3-s + 3/5·5-s − 3/7·7-s + 1/3·9-s − 1.63·11-s + 5/13·13-s + 3/5·15-s + 6/17·17-s − 3/7·21-s + 3.52·23-s + 1.71·25-s − 4/3·27-s + 2.37·29-s − 1.45·31-s − 1.63·33-s − 0.257·35-s − 0.540·37-s + 5/13·39-s + 1.31·41-s + 1/5·45-s − 4.40·47-s − 1.48·49-s + 6/17·51-s − 4.75·53-s − 0.981·55-s + 5.18·59-s + 7/61·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(56179.8\) |
| Root analytic conductor: | \(1.98083\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(3.054657995\) |
| \(L(\frac12)\) | \(\approx\) | \(3.054657995\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T + 2 p T^{2} + p^{3} T^{3} - 14 p^{2} T^{4} + p^{5} T^{5} + 2 p^{5} T^{6} - p^{7} T^{7} + p^{8} T^{8} \) | |
| good | 5 | \( 1 - 3 T - 34 T^{2} + 339 T^{3} + 211 T^{4} - 10296 T^{5} + 50708 T^{6} + 156576 T^{7} - 1757804 T^{8} + 156576 p^{2} T^{9} + 50708 p^{4} T^{10} - 10296 p^{6} T^{11} + 211 p^{8} T^{12} + 339 p^{10} T^{13} - 34 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 + 3 T + 82 T^{2} + 237 T^{3} + 2419 T^{4} + 21600 T^{5} + 116980 T^{6} + 2348256 T^{7} + 7641628 T^{8} + 2348256 p^{2} T^{9} + 116980 p^{4} T^{10} + 21600 p^{6} T^{11} + 2419 p^{8} T^{12} + 237 p^{10} T^{13} + 82 p^{12} T^{14} + 3 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 18 T + 448 T^{2} + 6120 T^{3} + 93463 T^{4} + 1253556 T^{5} + 16013428 T^{6} + 201806226 T^{7} + 2222076136 T^{8} + 201806226 p^{2} T^{9} + 16013428 p^{4} T^{10} + 1253556 p^{6} T^{11} + 93463 p^{8} T^{12} + 6120 p^{10} T^{13} + 448 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 - 5 T - 360 T^{2} - 1561 T^{3} + 77501 T^{4} + 597312 T^{5} - 5007086 T^{6} - 5241266 p T^{7} + 60552144 T^{8} - 5241266 p^{3} T^{9} - 5007086 p^{4} T^{10} + 597312 p^{6} T^{11} + 77501 p^{8} T^{12} - 1561 p^{10} T^{13} - 360 p^{12} T^{14} - 5 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( ( 1 - 3 T + 334 T^{2} - 693 T^{3} + 110178 T^{4} - 693 p^{2} T^{5} + 334 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 19 | \( 1 - 1157 T^{2} + 639226 T^{4} - 232952123 T^{6} + 79410905002 T^{8} - 232952123 p^{4} T^{10} + 639226 p^{8} T^{12} - 1157 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 - 81 T + 4870 T^{2} - 217323 T^{3} + 8414383 T^{4} - 278460504 T^{5} + 8325324592 T^{6} - 221238967356 T^{7} + 5380294547716 T^{8} - 221238967356 p^{2} T^{9} + 8325324592 p^{4} T^{10} - 278460504 p^{6} T^{11} + 8414383 p^{8} T^{12} - 217323 p^{10} T^{13} + 4870 p^{12} T^{14} - 81 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 69 T + 1976 T^{2} - 82905 T^{3} + 2728573 T^{4} - 24459264 T^{5} + 249830930 T^{6} + 10215187926 T^{7} - 1249058812400 T^{8} + 10215187926 p^{2} T^{9} + 249830930 p^{4} T^{10} - 24459264 p^{6} T^{11} + 2728573 p^{8} T^{12} - 82905 p^{10} T^{13} + 1976 p^{12} T^{14} - 69 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 + 45 T + 3520 T^{2} + 128025 T^{3} + 6239425 T^{4} + 221987952 T^{5} + 7979134318 T^{6} + 268603930890 T^{7} + 8040333650992 T^{8} + 268603930890 p^{2} T^{9} + 7979134318 p^{4} T^{10} + 221987952 p^{6} T^{11} + 6239425 p^{8} T^{12} + 128025 p^{10} T^{13} + 3520 p^{12} T^{14} + 45 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( ( 1 + 10 T + 1720 T^{2} - 76250 T^{3} + 347470 T^{4} - 76250 p^{2} T^{5} + 1720 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 41 | \( 1 - 54 T - 1108 T^{2} - 54648 T^{3} + 6254527 T^{4} + 135074304 T^{5} - 346116544 T^{6} - 221393044530 T^{7} - 6391866656792 T^{8} - 221393044530 p^{2} T^{9} - 346116544 p^{4} T^{10} + 135074304 p^{6} T^{11} + 6254527 p^{8} T^{12} - 54648 p^{10} T^{13} - 1108 p^{12} T^{14} - 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 4210 T^{2} + 9163465 T^{4} - 282271716 T^{5} + 12018514642 T^{6} - 1023475278060 T^{7} + 9810745543012 T^{8} - 1023475278060 p^{2} T^{9} + 12018514642 p^{4} T^{10} - 282271716 p^{6} T^{11} + 9163465 p^{8} T^{12} + 4210 p^{12} T^{14} + p^{16} T^{16} \) | |
| 47 | \( 1 + 207 T + 24406 T^{2} + 2095461 T^{3} + 142092151 T^{4} + 7940274480 T^{5} + 383592144688 T^{6} + 17253891231516 T^{7} + 786328390118740 T^{8} + 17253891231516 p^{2} T^{9} + 383592144688 p^{4} T^{10} + 7940274480 p^{6} T^{11} + 142092151 p^{8} T^{12} + 2095461 p^{10} T^{13} + 24406 p^{12} T^{14} + 207 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 + 126 T + 12208 T^{2} + 787698 T^{3} + 46295070 T^{4} + 787698 p^{2} T^{5} + 12208 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 - 306 T + 52768 T^{2} - 6596136 T^{3} + 655249567 T^{4} - 54840345300 T^{5} + 4031461665844 T^{6} - 267864455425266 T^{7} + 16404403358884120 T^{8} - 267864455425266 p^{2} T^{9} + 4031461665844 p^{4} T^{10} - 54840345300 p^{6} T^{11} + 655249567 p^{8} T^{12} - 6596136 p^{10} T^{13} + 52768 p^{12} T^{14} - 306 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 7 T - 8478 T^{2} - 347909 T^{3} + 38508983 T^{4} + 2257992504 T^{5} - 39583245248 T^{6} - 5257966541956 T^{7} - 30952228546836 T^{8} - 5257966541956 p^{2} T^{9} - 39583245248 p^{4} T^{10} + 2257992504 p^{6} T^{11} + 38508983 p^{8} T^{12} - 347909 p^{10} T^{13} - 8478 p^{12} T^{14} - 7 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 12 T + 11866 T^{2} + 141816 T^{3} + 66010801 T^{4} + 721613340 T^{5} + 400272803818 T^{6} + 3347530167216 T^{7} + 2268475959596500 T^{8} + 3347530167216 p^{2} T^{9} + 400272803818 p^{4} T^{10} + 721613340 p^{6} T^{11} + 66010801 p^{8} T^{12} + 141816 p^{10} T^{13} + 11866 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 14120 T^{2} + 150082588 T^{4} - 1080637518872 T^{6} + 6319440918414790 T^{8} - 1080637518872 p^{4} T^{10} + 150082588 p^{8} T^{12} - 14120 p^{12} T^{14} + p^{16} T^{16} \) | |
| 73 | \( ( 1 - 37 T + 20314 T^{2} - 565891 T^{3} + 160126666 T^{4} - 565891 p^{2} T^{5} + 20314 p^{4} T^{6} - 37 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 79 | \( 1 + 33 T + 13924 T^{2} + 447513 T^{3} + 83250565 T^{4} + 4145960160 T^{5} + 461050019242 T^{6} + 35083235957766 T^{7} + 3234248047096936 T^{8} + 35083235957766 p^{2} T^{9} + 461050019242 p^{4} T^{10} + 4145960160 p^{6} T^{11} + 83250565 p^{8} T^{12} + 447513 p^{10} T^{13} + 13924 p^{12} T^{14} + 33 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 + 549 T + 157876 T^{2} + 31517541 T^{3} + 4848813277 T^{4} + 610140548160 T^{5} + 65668976887426 T^{6} + 6260783050033902 T^{7} + 542374558465652968 T^{8} + 6260783050033902 p^{2} T^{9} + 65668976887426 p^{4} T^{10} + 610140548160 p^{6} T^{11} + 4848813277 p^{8} T^{12} + 31517541 p^{10} T^{13} + 157876 p^{12} T^{14} + 549 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( ( 1 + 84 T + 30700 T^{2} + 1886940 T^{3} + 359702982 T^{4} + 1886940 p^{2} T^{5} + 30700 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 97 | \( 1 + 10 T - 22548 T^{2} + 828296 T^{3} + 272325791 T^{4} - 14518511616 T^{5} - 1388516520704 T^{6} + 83318070893038 T^{7} + 5416086623373288 T^{8} + 83318070893038 p^{2} T^{9} - 1388516520704 p^{4} T^{10} - 14518511616 p^{6} T^{11} + 272325791 p^{8} T^{12} + 828296 p^{10} T^{13} - 22548 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−5.76263742681022596609106488481, −5.50474316594636149366239532205, −5.42075684438861305504740391209, −5.41525695758708801637139498472, −5.05041716409600772187036514376, −4.95330425603699905711625304301, −4.80089977219270124886536227260, −4.70855749121947442517902301779, −4.45045011868136400471517243094, −4.24711781091870754236915002742, −4.07623254617083130457941203306, −3.86363145228645716175168258560, −3.32373550146044868145890911951, −3.21821565730411576244851481544, −3.02185421401737778103506941292, −3.01659970007905456229927701724, −3.01493985709193450447813414433, −3.01111631675284774800348906146, −2.22258552892502357342704407724, −2.15315996536543920158095248299, −1.91727755357697426061247161673, −1.50359759019870865259166482133, −1.24019238449687521463350491646, −0.897722648654943688774459832613, −0.27976211106785359599932910644, 0.27976211106785359599932910644, 0.897722648654943688774459832613, 1.24019238449687521463350491646, 1.50359759019870865259166482133, 1.91727755357697426061247161673, 2.15315996536543920158095248299, 2.22258552892502357342704407724, 3.01111631675284774800348906146, 3.01493985709193450447813414433, 3.01659970007905456229927701724, 3.02185421401737778103506941292, 3.21821565730411576244851481544, 3.32373550146044868145890911951, 3.86363145228645716175168258560, 4.07623254617083130457941203306, 4.24711781091870754236915002742, 4.45045011868136400471517243094, 4.70855749121947442517902301779, 4.80089977219270124886536227260, 4.95330425603699905711625304301, 5.05041716409600772187036514376, 5.41525695758708801637139498472, 5.42075684438861305504740391209, 5.50474316594636149366239532205, 5.76263742681022596609106488481
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.