Dirichlet series
| L(s) = 1 | + 2·5-s − 12·7-s + 7·11-s − 9·13-s + 17-s − 11·19-s + 19·23-s − 14·25-s + 5·29-s − 13·31-s − 24·35-s − 26·37-s + 2·41-s − 24·43-s + 11·47-s + 57·49-s − 4·53-s + 14·55-s + 4·59-s − 5·61-s − 18·65-s − 42·67-s − 9·71-s + 27·73-s − 84·77-s − 8·79-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 4.53·7-s + 2.11·11-s − 2.49·13-s + 0.242·17-s − 2.52·19-s + 3.96·23-s − 2.79·25-s + 0.928·29-s − 2.33·31-s − 4.05·35-s − 4.27·37-s + 0.312·41-s − 3.65·43-s + 1.60·47-s + 57/7·49-s − 0.549·53-s + 1.88·55-s + 0.520·59-s − 0.640·61-s − 2.23·65-s − 5.13·67-s − 1.06·71-s + 3.16·73-s − 9.57·77-s − 0.900·79-s − 1.75·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 3^{14} \cdot 167^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(5.87587\times 10^{11}\) |
| Root analytic conductor: | \(6.92864\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{14} \cdot 3^{14} \cdot 167^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | \( 1 \) |
| 3 | \( 1 \) | |
| 167 | \( ( 1 + T )^{7} \) | |
| good | 5 | \( 1 - 2 T + 18 T^{2} - 18 T^{3} + 128 T^{4} + 2 T^{5} + 109 p T^{6} + 452 T^{7} + 109 p^{2} T^{8} + 2 p^{2} T^{9} + 128 p^{3} T^{10} - 18 p^{4} T^{11} + 18 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + 12 T + 87 T^{2} + 466 T^{3} + 2053 T^{4} + 7675 T^{5} + 3547 p T^{6} + 70131 T^{7} + 3547 p^{2} T^{8} + 7675 p^{2} T^{9} + 2053 p^{3} T^{10} + 466 p^{4} T^{11} + 87 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 7 T + 49 T^{2} - 211 T^{3} + 1044 T^{4} - 3820 T^{5} + 15876 T^{6} - 49868 T^{7} + 15876 p T^{8} - 3820 p^{2} T^{9} + 1044 p^{3} T^{10} - 211 p^{4} T^{11} + 49 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 9 T + 68 T^{2} + 356 T^{3} + 1658 T^{4} + 6631 T^{5} + 26213 T^{6} + 92808 T^{7} + 26213 p T^{8} + 6631 p^{2} T^{9} + 1658 p^{3} T^{10} + 356 p^{4} T^{11} + 68 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - T + 66 T^{2} - 26 T^{3} + 2212 T^{4} - 199 T^{5} + 51705 T^{6} - 1532 T^{7} + 51705 p T^{8} - 199 p^{2} T^{9} + 2212 p^{3} T^{10} - 26 p^{4} T^{11} + 66 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 11 T + 123 T^{2} + 943 T^{3} + 6780 T^{4} + 38558 T^{5} + 207830 T^{6} + 937252 T^{7} + 207830 p T^{8} + 38558 p^{2} T^{9} + 6780 p^{3} T^{10} + 943 p^{4} T^{11} + 123 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 19 T + 232 T^{2} - 2074 T^{3} + 15558 T^{4} - 100741 T^{5} + 582135 T^{6} - 2958428 T^{7} + 582135 p T^{8} - 100741 p^{2} T^{9} + 15558 p^{3} T^{10} - 2074 p^{4} T^{11} + 232 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 5 T + 162 T^{2} - 702 T^{3} + 12203 T^{4} - 45007 T^{5} + 550271 T^{6} - 1671295 T^{7} + 550271 p T^{8} - 45007 p^{2} T^{9} + 12203 p^{3} T^{10} - 702 p^{4} T^{11} + 162 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 13 T + 95 T^{2} + 305 T^{3} + 648 T^{4} - 560 T^{5} + 20132 T^{6} + 105568 T^{7} + 20132 p T^{8} - 560 p^{2} T^{9} + 648 p^{3} T^{10} + 305 p^{4} T^{11} + 95 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 26 T + 404 T^{2} + 4432 T^{3} + 39662 T^{4} + 308446 T^{5} + 2171097 T^{6} + 13871632 T^{7} + 2171097 p T^{8} + 308446 p^{2} T^{9} + 39662 p^{3} T^{10} + 4432 p^{4} T^{11} + 404 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 2 T + 146 T^{2} - 266 T^{3} + 11268 T^{4} - 526 p T^{5} + 619985 T^{6} - 1112908 T^{7} + 619985 p T^{8} - 526 p^{3} T^{9} + 11268 p^{3} T^{10} - 266 p^{4} T^{11} + 146 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 24 T + 514 T^{2} + 7062 T^{3} + 86260 T^{4} + 816360 T^{5} + 6931999 T^{6} + 47907988 T^{7} + 6931999 p T^{8} + 816360 p^{2} T^{9} + 86260 p^{3} T^{10} + 7062 p^{4} T^{11} + 514 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 11 T + 154 T^{2} - 1124 T^{3} + 10727 T^{4} - 58321 T^{5} + 473997 T^{6} - 2317571 T^{7} + 473997 p T^{8} - 58321 p^{2} T^{9} + 10727 p^{3} T^{10} - 1124 p^{4} T^{11} + 154 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 4 T + 207 T^{2} + 920 T^{3} + 21673 T^{4} + 86044 T^{5} + 1568415 T^{6} + 5154512 T^{7} + 1568415 p T^{8} + 86044 p^{2} T^{9} + 21673 p^{3} T^{10} + 920 p^{4} T^{11} + 207 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 4 T + 321 T^{2} - 16 p T^{3} + 46921 T^{4} - 103612 T^{5} + 4159569 T^{6} - 7292384 T^{7} + 4159569 p T^{8} - 103612 p^{2} T^{9} + 46921 p^{3} T^{10} - 16 p^{5} T^{11} + 321 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 5 T + 210 T^{2} + 398 T^{3} + 22363 T^{4} + 26651 T^{5} + 1945003 T^{6} + 2595919 T^{7} + 1945003 p T^{8} + 26651 p^{2} T^{9} + 22363 p^{3} T^{10} + 398 p^{4} T^{11} + 210 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 42 T + 1060 T^{2} + 19258 T^{3} + 278042 T^{4} + 3324230 T^{5} + 33867347 T^{6} + 297649804 T^{7} + 33867347 p T^{8} + 3324230 p^{2} T^{9} + 278042 p^{3} T^{10} + 19258 p^{4} T^{11} + 1060 p^{5} T^{12} + 42 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 9 T + 320 T^{2} + 2246 T^{3} + 49410 T^{4} + 293863 T^{5} + 5051659 T^{6} + 25462260 T^{7} + 5051659 p T^{8} + 293863 p^{2} T^{9} + 49410 p^{3} T^{10} + 2246 p^{4} T^{11} + 320 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 27 T + 614 T^{2} - 10216 T^{3} + 143660 T^{4} - 1707773 T^{5} + 17858693 T^{6} - 161388576 T^{7} + 17858693 p T^{8} - 1707773 p^{2} T^{9} + 143660 p^{3} T^{10} - 10216 p^{4} T^{11} + 614 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 8 T + 314 T^{2} + 2854 T^{3} + 52244 T^{4} + 477640 T^{5} + 5822187 T^{6} + 47351668 T^{7} + 5822187 p T^{8} + 477640 p^{2} T^{9} + 52244 p^{3} T^{10} + 2854 p^{4} T^{11} + 314 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 16 T + 238 T^{2} + 22 p T^{3} + 19504 T^{4} + 186096 T^{5} + 2712391 T^{6} + 24177788 T^{7} + 2712391 p T^{8} + 186096 p^{2} T^{9} + 19504 p^{3} T^{10} + 22 p^{5} T^{11} + 238 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 9 T + 261 T^{2} + 1817 T^{3} + 33900 T^{4} + 261500 T^{5} + 47480 p T^{6} + 30821292 T^{7} + 47480 p^{2} T^{8} + 261500 p^{2} T^{9} + 33900 p^{3} T^{10} + 1817 p^{4} T^{11} + 261 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 8 T + 417 T^{2} + 3412 T^{3} + 94457 T^{4} + 671935 T^{5} + 13479577 T^{6} + 82425871 T^{7} + 13479577 p T^{8} + 671935 p^{2} T^{9} + 94457 p^{3} T^{10} + 3412 p^{4} T^{11} + 417 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.92861921491439427002470803397, −3.89551732403664072888841003690, −3.87305742042791300550412101169, −3.81926184340911232241354506893, −3.50440826776738804488333113848, −3.40412343264478508805053917765, −3.27493169384653901231368182405, −3.24509010839170528213229960325, −3.19717335609936056287869468766, −3.00866898149550531069517470321, −2.94459815402758536067438093740, −2.65715898522255319890867097906, −2.59018130736814974370893549936, −2.58699909596995142104094880282, −2.33802060552139830128003764472, −2.21849668295077873425536057509, −2.16919090028782273055818875306, −2.11412541612578460480467550203, −1.61708716739617398299809786594, −1.55439048629871084615670487931, −1.43161566733272371642327798372, −1.40891156435649090877202780522, −1.28595055111867425313929982183, −1.12061083300263257502009643979, −0.964203304489066089364547830646, 0, 0, 0, 0, 0, 0, 0, 0.964203304489066089364547830646, 1.12061083300263257502009643979, 1.28595055111867425313929982183, 1.40891156435649090877202780522, 1.43161566733272371642327798372, 1.55439048629871084615670487931, 1.61708716739617398299809786594, 2.11412541612578460480467550203, 2.16919090028782273055818875306, 2.21849668295077873425536057509, 2.33802060552139830128003764472, 2.58699909596995142104094880282, 2.59018130736814974370893549936, 2.65715898522255319890867097906, 2.94459815402758536067438093740, 3.00866898149550531069517470321, 3.19717335609936056287869468766, 3.24509010839170528213229960325, 3.27493169384653901231368182405, 3.40412343264478508805053917765, 3.50440826776738804488333113848, 3.81926184340911232241354506893, 3.87305742042791300550412101169, 3.89551732403664072888841003690, 3.92861921491439427002470803397
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.