Dirichlet series
| L(s) = 1 | − 2·3-s + 2·5-s − 10·7-s − 3·9-s + 7·11-s − 4·13-s − 4·15-s + 2·17-s − 7·19-s + 20·21-s − 10·23-s − 11·25-s + 12·27-s − 18·29-s − 24·31-s − 14·33-s − 20·35-s + 8·39-s − 12·41-s − 2·43-s − 6·45-s − 8·47-s + 34·49-s − 4·51-s + 2·53-s + 14·55-s + 14·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 3.77·7-s − 9-s + 2.11·11-s − 1.10·13-s − 1.03·15-s + 0.485·17-s − 1.60·19-s + 4.36·21-s − 2.08·23-s − 2.19·25-s + 2.30·27-s − 3.34·29-s − 4.31·31-s − 2.43·33-s − 3.38·35-s + 1.28·39-s − 1.87·41-s − 0.304·43-s − 0.894·45-s − 1.16·47-s + 34/7·49-s − 0.560·51-s + 0.274·53-s + 1.88·55-s + 1.85·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{28} \cdot 11^{7} \cdot 19^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(9.67840\times 10^{9}\) |
| Root analytic conductor: | \(5.16739\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{28} \cdot 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( ( 1 - T )^{7} \) | |
| 19 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 + 2 T + 7 T^{2} + 8 T^{3} + 25 T^{4} + 34 T^{5} + 82 T^{6} + 104 T^{7} + 82 p T^{8} + 34 p^{2} T^{9} + 25 p^{3} T^{10} + 8 p^{4} T^{11} + 7 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - 2 T + 3 p T^{2} - 26 T^{3} + 113 T^{4} - 226 T^{5} + 752 T^{6} - 1466 T^{7} + 752 p T^{8} - 226 p^{2} T^{9} + 113 p^{3} T^{10} - 26 p^{4} T^{11} + 3 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 10 T + 66 T^{2} + 334 T^{3} + 1439 T^{4} + 5258 T^{5} + 16844 T^{6} + 47228 T^{7} + 16844 p T^{8} + 5258 p^{2} T^{9} + 1439 p^{3} T^{10} + 334 p^{4} T^{11} + 66 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 4 T + 40 T^{2} + 118 T^{3} + 873 T^{4} + 2134 T^{5} + 13076 T^{6} + 27460 T^{7} + 13076 p T^{8} + 2134 p^{2} T^{9} + 873 p^{3} T^{10} + 118 p^{4} T^{11} + 40 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 2 T + 49 T^{2} - 160 T^{3} + 1671 T^{4} - 4814 T^{5} + 2199 p T^{6} - 107936 T^{7} + 2199 p^{2} T^{8} - 4814 p^{2} T^{9} + 1671 p^{3} T^{10} - 160 p^{4} T^{11} + 49 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 10 T + 110 T^{2} + 732 T^{3} + 4928 T^{4} + 24870 T^{5} + 5981 p T^{6} + 610984 T^{7} + 5981 p^{2} T^{8} + 24870 p^{2} T^{9} + 4928 p^{3} T^{10} + 732 p^{4} T^{11} + 110 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 18 T + 320 T^{2} + 3472 T^{3} + 35009 T^{4} + 266396 T^{5} + 1870372 T^{6} + 10488792 T^{7} + 1870372 p T^{8} + 266396 p^{2} T^{9} + 35009 p^{3} T^{10} + 3472 p^{4} T^{11} + 320 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 24 T + 431 T^{2} + 5368 T^{3} + 55269 T^{4} + 459990 T^{5} + 3278314 T^{6} + 19632048 T^{7} + 3278314 p T^{8} + 459990 p^{2} T^{9} + 55269 p^{3} T^{10} + 5368 p^{4} T^{11} + 431 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 138 T^{2} - 194 T^{3} + 9876 T^{4} - 19416 T^{5} + 504517 T^{6} - 914604 T^{7} + 504517 p T^{8} - 19416 p^{2} T^{9} + 9876 p^{3} T^{10} - 194 p^{4} T^{11} + 138 p^{5} T^{12} + p^{7} T^{14} \) | |
| 41 | \( 1 + 12 T + 282 T^{2} + 2426 T^{3} + 32453 T^{4} + 216142 T^{5} + 2107796 T^{6} + 11219712 T^{7} + 2107796 p T^{8} + 216142 p^{2} T^{9} + 32453 p^{3} T^{10} + 2426 p^{4} T^{11} + 282 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 2 T + 212 T^{2} + 366 T^{3} + 497 p T^{4} + 30878 T^{5} + 1346480 T^{6} + 1615092 T^{7} + 1346480 p T^{8} + 30878 p^{2} T^{9} + 497 p^{4} T^{10} + 366 p^{4} T^{11} + 212 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 8 T + 177 T^{2} + 912 T^{3} + 11693 T^{4} + 35256 T^{5} + 432605 T^{6} + 866144 T^{7} + 432605 p T^{8} + 35256 p^{2} T^{9} + 11693 p^{3} T^{10} + 912 p^{4} T^{11} + 177 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 2 T + 211 T^{2} - 604 T^{3} + 22621 T^{4} - 63566 T^{5} + 1684215 T^{6} - 3939464 T^{7} + 1684215 p T^{8} - 63566 p^{2} T^{9} + 22621 p^{3} T^{10} - 604 p^{4} T^{11} + 211 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 10 T + 68 T^{2} - 564 T^{3} + 7490 T^{4} - 69606 T^{5} + 453907 T^{6} - 1842328 T^{7} + 453907 p T^{8} - 69606 p^{2} T^{9} + 7490 p^{3} T^{10} - 564 p^{4} T^{11} + 68 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 14 T + 393 T^{2} - 4080 T^{3} + 66043 T^{4} - 544594 T^{5} + 105303 p T^{6} - 42469120 T^{7} + 105303 p^{2} T^{8} - 544594 p^{2} T^{9} + 66043 p^{3} T^{10} - 4080 p^{4} T^{11} + 393 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 8 T + 299 T^{2} + 1908 T^{3} + 43661 T^{4} + 221222 T^{5} + 4054526 T^{6} + 17312388 T^{7} + 4054526 p T^{8} + 221222 p^{2} T^{9} + 43661 p^{3} T^{10} + 1908 p^{4} T^{11} + 299 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 10 T + 363 T^{2} + 3316 T^{3} + 60569 T^{4} + 7042 p T^{5} + 6248102 T^{6} + 44683996 T^{7} + 6248102 p T^{8} + 7042 p^{3} T^{9} + 60569 p^{3} T^{10} + 3316 p^{4} T^{11} + 363 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 6 T + 291 T^{2} + 1036 T^{3} + 35145 T^{4} + 59322 T^{5} + 2766403 T^{6} + 2354856 T^{7} + 2766403 p T^{8} + 59322 p^{2} T^{9} + 35145 p^{3} T^{10} + 1036 p^{4} T^{11} + 291 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 52 T + 1523 T^{2} + 31800 T^{3} + 522203 T^{4} + 7037132 T^{5} + 79661705 T^{6} + 766418576 T^{7} + 79661705 p T^{8} + 7037132 p^{2} T^{9} + 522203 p^{3} T^{10} + 31800 p^{4} T^{11} + 1523 p^{5} T^{12} + 52 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 10 T + 362 T^{2} - 1618 T^{3} + 45511 T^{4} + 11482 T^{5} + 3276004 T^{6} + 12186140 T^{7} + 3276004 p T^{8} + 11482 p^{2} T^{9} + 45511 p^{3} T^{10} - 1618 p^{4} T^{11} + 362 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 222 T^{2} - 698 T^{3} + 38288 T^{4} - 87304 T^{5} + 4414265 T^{6} - 12681948 T^{7} + 4414265 p T^{8} - 87304 p^{2} T^{9} + 38288 p^{3} T^{10} - 698 p^{4} T^{11} + 222 p^{5} T^{12} + p^{7} T^{14} \) | |
| 97 | \( 1 + 24 T + 490 T^{2} + 7290 T^{3} + 114080 T^{4} + 1401624 T^{5} + 16484869 T^{6} + 161147084 T^{7} + 16484869 p T^{8} + 1401624 p^{2} T^{9} + 114080 p^{3} T^{10} + 7290 p^{4} T^{11} + 490 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.29874566428575760219475594647, −4.05352745777888575307352246959, −3.96876363645154699780832653829, −3.95450787665830360366217154545, −3.86251741941895803089638472868, −3.71425661888707943068421178394, −3.60044522559543775777636633338, −3.54599696305820065513393538169, −3.53912105631211842343381179690, −3.44933633953984393212233351301, −3.20717349582467339245621173779, −2.81686904548712357082175024326, −2.73816446387778668869635994370, −2.66357623335323196513486484207, −2.60896616369521154740192171692, −2.56504232593657570894048958661, −2.23779754804505208772287391112, −2.17224957713772946388474040427, −1.97670071013905631039505922227, −1.67666416403526608824822246126, −1.55184711201661802518885352712, −1.48995287513377870948440849954, −1.46153496748798818328200574360, −1.30164856754113484809788743669, −0.977762279250492598084285917564, 0, 0, 0, 0, 0, 0, 0, 0.977762279250492598084285917564, 1.30164856754113484809788743669, 1.46153496748798818328200574360, 1.48995287513377870948440849954, 1.55184711201661802518885352712, 1.67666416403526608824822246126, 1.97670071013905631039505922227, 2.17224957713772946388474040427, 2.23779754804505208772287391112, 2.56504232593657570894048958661, 2.60896616369521154740192171692, 2.66357623335323196513486484207, 2.73816446387778668869635994370, 2.81686904548712357082175024326, 3.20717349582467339245621173779, 3.44933633953984393212233351301, 3.53912105631211842343381179690, 3.54599696305820065513393538169, 3.60044522559543775777636633338, 3.71425661888707943068421178394, 3.86251741941895803089638472868, 3.95450787665830360366217154545, 3.96876363645154699780832653829, 4.05352745777888575307352246959, 4.29874566428575760219475594647
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.