Dirichlet series
| L(s) = 1 | − 7·3-s − 3·5-s + 8·7-s + 28·9-s + 11-s − 2·13-s + 21·15-s + 11·17-s + 2·19-s − 56·21-s + 17·23-s − 11·25-s − 84·27-s − 7·29-s + 10·31-s − 7·33-s − 24·35-s − 21·37-s + 14·39-s + 8·41-s − 12·43-s − 84·45-s + 25·47-s + 4·49-s − 77·51-s − 7·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 4.04·3-s − 1.34·5-s + 3.02·7-s + 28/3·9-s + 0.301·11-s − 0.554·13-s + 5.42·15-s + 2.66·17-s + 0.458·19-s − 12.2·21-s + 3.54·23-s − 2.19·25-s − 16.1·27-s − 1.29·29-s + 1.79·31-s − 1.21·33-s − 4.05·35-s − 3.45·37-s + 2.24·39-s + 1.24·41-s − 1.82·43-s − 12.5·45-s + 3.64·47-s + 4/7·49-s − 10.7·51-s − 0.961·53-s − 0.404·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{7} \cdot 167^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 3^{7} \cdot 167^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.43901\times 10^{10}\) |
| Root analytic conductor: | \(5.65721\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{21} \cdot 3^{7} \cdot 167^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.082044857\) |
| \(L(\frac12)\) | \(\approx\) | \(3.082044857\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T )^{7} \) | |
| 167 | \( ( 1 + T )^{7} \) | |
| good | 5 | \( 1 + 3 T + 4 p T^{2} + 8 p T^{3} + 173 T^{4} + 258 T^{5} + 1026 T^{6} + 1326 T^{7} + 1026 p T^{8} + 258 p^{2} T^{9} + 173 p^{3} T^{10} + 8 p^{5} T^{11} + 4 p^{6} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 8 T + 60 T^{2} - 281 T^{3} + 181 p T^{4} - 4377 T^{5} + 2092 p T^{6} - 39420 T^{7} + 2092 p^{2} T^{8} - 4377 p^{2} T^{9} + 181 p^{4} T^{10} - 281 p^{4} T^{11} + 60 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - T + 46 T^{2} - 28 T^{3} + 1061 T^{4} - 414 T^{5} + 16252 T^{6} - 5010 T^{7} + 16252 p T^{8} - 414 p^{2} T^{9} + 1061 p^{3} T^{10} - 28 p^{4} T^{11} + 46 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 2 T + 48 T^{2} + 20 T^{3} + 854 T^{4} - 1667 T^{5} + 8225 T^{6} - 41310 T^{7} + 8225 p T^{8} - 1667 p^{2} T^{9} + 854 p^{3} T^{10} + 20 p^{4} T^{11} + 48 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 11 T + 125 T^{2} - 904 T^{3} + 6292 T^{4} - 34384 T^{5} + 176026 T^{6} - 750594 T^{7} + 176026 p T^{8} - 34384 p^{2} T^{9} + 6292 p^{3} T^{10} - 904 p^{4} T^{11} + 125 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 2 T + 68 T^{2} - 280 T^{3} + 2522 T^{4} - 11611 T^{5} + 70571 T^{6} - 268206 T^{7} + 70571 p T^{8} - 11611 p^{2} T^{9} + 2522 p^{3} T^{10} - 280 p^{4} T^{11} + 68 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 17 T + 264 T^{2} - 2580 T^{3} + 22869 T^{4} - 155172 T^{5} + 962930 T^{6} - 4821022 T^{7} + 962930 p T^{8} - 155172 p^{2} T^{9} + 22869 p^{3} T^{10} - 2580 p^{4} T^{11} + 264 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 7 T + 146 T^{2} + 790 T^{3} + 9969 T^{4} + 44070 T^{5} + 422776 T^{6} + 1553690 T^{7} + 422776 p T^{8} + 44070 p^{2} T^{9} + 9969 p^{3} T^{10} + 790 p^{4} T^{11} + 146 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 10 T + 211 T^{2} - 1575 T^{3} + 18783 T^{4} - 110477 T^{5} + 946125 T^{6} - 4420900 T^{7} + 946125 p T^{8} - 110477 p^{2} T^{9} + 18783 p^{3} T^{10} - 1575 p^{4} T^{11} + 211 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 21 T + 360 T^{2} + 4288 T^{3} + 43987 T^{4} + 370784 T^{5} + 2772232 T^{6} + 17825150 T^{7} + 2772232 p T^{8} + 370784 p^{2} T^{9} + 43987 p^{3} T^{10} + 4288 p^{4} T^{11} + 360 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 8 T + 5 p T^{2} - 1417 T^{3} + 20339 T^{4} - 118045 T^{5} + 1250235 T^{6} - 6008956 T^{7} + 1250235 p T^{8} - 118045 p^{2} T^{9} + 20339 p^{3} T^{10} - 1417 p^{4} T^{11} + 5 p^{6} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 12 T + 145 T^{2} + 23 p T^{3} + 7307 T^{4} + 31281 T^{5} + 187515 T^{6} + 541084 T^{7} + 187515 p T^{8} + 31281 p^{2} T^{9} + 7307 p^{3} T^{10} + 23 p^{5} T^{11} + 145 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 25 T + 440 T^{2} - 5547 T^{3} + 60316 T^{4} - 549620 T^{5} + 4505677 T^{6} - 32294448 T^{7} + 4505677 p T^{8} - 549620 p^{2} T^{9} + 60316 p^{3} T^{10} - 5547 p^{4} T^{11} + 440 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 7 T + 166 T^{2} + 1295 T^{3} + 19306 T^{4} + 115366 T^{5} + 1391231 T^{6} + 7778352 T^{7} + 1391231 p T^{8} + 115366 p^{2} T^{9} + 19306 p^{3} T^{10} + 1295 p^{4} T^{11} + 166 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 3 T + 362 T^{2} - 997 T^{3} + 58918 T^{4} - 141144 T^{5} + 5562009 T^{6} - 10948080 T^{7} + 5562009 p T^{8} - 141144 p^{2} T^{9} + 58918 p^{3} T^{10} - 997 p^{4} T^{11} + 362 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 14 T + 304 T^{2} + 3390 T^{3} + 45370 T^{4} + 399265 T^{5} + 4077845 T^{6} + 29981926 T^{7} + 4077845 p T^{8} + 399265 p^{2} T^{9} + 45370 p^{3} T^{10} + 3390 p^{4} T^{11} + 304 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 382 T^{2} - 1324 T^{3} + 66982 T^{4} - 198497 T^{5} + 6988981 T^{6} - 17114094 T^{7} + 6988981 p T^{8} - 198497 p^{2} T^{9} + 66982 p^{3} T^{10} - 1324 p^{4} T^{11} + 382 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 27 T + 731 T^{2} - 12149 T^{3} + 188468 T^{4} - 2220242 T^{5} + 24216158 T^{6} - 212111532 T^{7} + 24216158 p T^{8} - 2220242 p^{2} T^{9} + 188468 p^{3} T^{10} - 12149 p^{4} T^{11} + 731 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 12 T + 286 T^{2} + 1814 T^{3} + 32184 T^{4} + 167073 T^{5} + 3162525 T^{6} + 15750450 T^{7} + 3162525 p T^{8} + 167073 p^{2} T^{9} + 32184 p^{3} T^{10} + 1814 p^{4} T^{11} + 286 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 8 T + 463 T^{2} - 2721 T^{3} + 93163 T^{4} - 415133 T^{5} + 11094341 T^{6} - 39502788 T^{7} + 11094341 p T^{8} - 415133 p^{2} T^{9} + 93163 p^{3} T^{10} - 2721 p^{4} T^{11} + 463 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 15 T + 252 T^{2} - 2335 T^{3} + 40204 T^{4} - 384018 T^{5} + 4378733 T^{6} - 30821664 T^{7} + 4378733 p T^{8} - 384018 p^{2} T^{9} + 40204 p^{3} T^{10} - 2335 p^{4} T^{11} + 252 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 14 T + 532 T^{2} - 6878 T^{3} + 130172 T^{4} - 1444175 T^{5} + 18617595 T^{6} - 167849250 T^{7} + 18617595 p T^{8} - 1444175 p^{2} T^{9} + 130172 p^{3} T^{10} - 6878 p^{4} T^{11} + 532 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 3 T + 475 T^{2} - 1207 T^{3} + 104508 T^{4} - 227430 T^{5} + 14447348 T^{6} - 26865048 T^{7} + 14447348 p T^{8} - 227430 p^{2} T^{9} + 104508 p^{3} T^{10} - 1207 p^{4} T^{11} + 475 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.93136227900958104740856282958, −3.92451311055458877585008007118, −3.80956486970616755247145068805, −3.64466482481077932414300845478, −3.30768278110998567997457707315, −3.27399015798948725841846118670, −3.21500529053094666054560455777, −2.97880269554946201719042615092, −2.78568321783171688783345902309, −2.77788761258639635616981621206, −2.76104468240840480031698080852, −2.05768441220548396916584711178, −2.02704889271303671908603000836, −1.90516489149977416127682917228, −1.87686017109153243897704234037, −1.67429047556575947667269673388, −1.64037572365839893412781602268, −1.56126527380372201695552480189, −1.21087594898677244403285284082, −1.01052976772749059155210914057, −0.880374650197562619114921898332, −0.77398027080403547007068129865, −0.62120430203534693399459627492, −0.40012813835925092910818212629, −0.28395693868843407814609047578, 0.28395693868843407814609047578, 0.40012813835925092910818212629, 0.62120430203534693399459627492, 0.77398027080403547007068129865, 0.880374650197562619114921898332, 1.01052976772749059155210914057, 1.21087594898677244403285284082, 1.56126527380372201695552480189, 1.64037572365839893412781602268, 1.67429047556575947667269673388, 1.87686017109153243897704234037, 1.90516489149977416127682917228, 2.02704889271303671908603000836, 2.05768441220548396916584711178, 2.76104468240840480031698080852, 2.77788761258639635616981621206, 2.78568321783171688783345902309, 2.97880269554946201719042615092, 3.21500529053094666054560455777, 3.27399015798948725841846118670, 3.30768278110998567997457707315, 3.64466482481077932414300845478, 3.80956486970616755247145068805, 3.92451311055458877585008007118, 3.93136227900958104740856282958
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.