Dirichlet series
| L(s) = 1 | + 3-s − 2·5-s + 7·7-s − 2·9-s + 11-s − 2·15-s − 5·17-s + 19-s + 7·21-s − 2·23-s − 7·25-s + 4·27-s + 4·29-s + 10·31-s + 33-s − 14·35-s − 5·37-s − 7·41-s − 7·43-s + 4·45-s + 10·47-s + 28·49-s − 5·51-s + 13·53-s − 2·55-s + 57-s + 2·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 2.64·7-s − 2/3·9-s + 0.301·11-s − 0.516·15-s − 1.21·17-s + 0.229·19-s + 1.52·21-s − 0.417·23-s − 7/5·25-s + 0.769·27-s + 0.742·29-s + 1.79·31-s + 0.174·33-s − 2.36·35-s − 0.821·37-s − 1.09·41-s − 1.06·43-s + 0.596·45-s + 1.45·47-s + 4·49-s − 0.700·51-s + 1.78·53-s − 0.269·55-s + 0.132·57-s + 0.260·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 7^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{21} \cdot 7^{7} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.40754\times 10^{13}\) |
| Root analytic conductor: | \(8.69312\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{21} \cdot 7^{7} \cdot 13^{14} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(28.01790332\) |
| \(L(\frac12)\) | \(\approx\) | \(28.01790332\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - T )^{7} \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} - p^{2} T^{3} + 14 T^{4} - 7 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + 2 p^{4} T^{8} - 7 p^{3} T^{9} + 14 p^{3} T^{10} - p^{6} T^{11} + p^{6} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 2 T + 11 T^{2} + 41 T^{3} + 126 T^{4} + 322 T^{5} + 176 p T^{6} + 2091 T^{7} + 176 p^{2} T^{8} + 322 p^{2} T^{9} + 126 p^{3} T^{10} + 41 p^{4} T^{11} + 11 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - T + 12 T^{2} - 36 T^{3} + 244 T^{4} - 523 T^{5} + 3517 T^{6} - 5872 T^{7} + 3517 p T^{8} - 523 p^{2} T^{9} + 244 p^{3} T^{10} - 36 p^{4} T^{11} + 12 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 5 T + 63 T^{2} + 166 T^{3} + 1704 T^{4} + 3650 T^{5} + 40064 T^{6} + 82334 T^{7} + 40064 p T^{8} + 3650 p^{2} T^{9} + 1704 p^{3} T^{10} + 166 p^{4} T^{11} + 63 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - T + 86 T^{2} - 138 T^{3} + 3620 T^{4} - 6775 T^{5} + 97859 T^{6} - 172428 T^{7} + 97859 p T^{8} - 6775 p^{2} T^{9} + 3620 p^{3} T^{10} - 138 p^{4} T^{11} + 86 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 2 T + 28 T^{2} + 64 T^{3} + 316 T^{4} - 1630 T^{5} + 201 T^{6} - 102616 T^{7} + 201 p T^{8} - 1630 p^{2} T^{9} + 316 p^{3} T^{10} + 64 p^{4} T^{11} + 28 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 4 T + 92 T^{2} - 495 T^{3} + 5432 T^{4} - 25425 T^{5} + 223819 T^{6} - 880312 T^{7} + 223819 p T^{8} - 25425 p^{2} T^{9} + 5432 p^{3} T^{10} - 495 p^{4} T^{11} + 92 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 10 T + 160 T^{2} - 1032 T^{3} + 10256 T^{4} - 52074 T^{5} + 424657 T^{6} - 1855704 T^{7} + 424657 p T^{8} - 52074 p^{2} T^{9} + 10256 p^{3} T^{10} - 1032 p^{4} T^{11} + 160 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 5 T + 110 T^{2} + 665 T^{3} + 6708 T^{4} + 35895 T^{5} + 313249 T^{6} + 1362086 T^{7} + 313249 p T^{8} + 35895 p^{2} T^{9} + 6708 p^{3} T^{10} + 665 p^{4} T^{11} + 110 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 7 T + 106 T^{2} + 985 T^{3} + 10290 T^{4} + 67785 T^{5} + 566179 T^{6} + 89502 p T^{7} + 566179 p T^{8} + 67785 p^{2} T^{9} + 10290 p^{3} T^{10} + 985 p^{4} T^{11} + 106 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 7 T + 170 T^{2} + 1488 T^{3} + 16802 T^{4} + 127549 T^{5} + 1132709 T^{6} + 6629928 T^{7} + 1132709 p T^{8} + 127549 p^{2} T^{9} + 16802 p^{3} T^{10} + 1488 p^{4} T^{11} + 170 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 10 T + 246 T^{2} - 1812 T^{3} + 26188 T^{4} - 154790 T^{5} + 1718215 T^{6} - 8582072 T^{7} + 1718215 p T^{8} - 154790 p^{2} T^{9} + 26188 p^{3} T^{10} - 1812 p^{4} T^{11} + 246 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 13 T + 301 T^{2} - 2312 T^{3} + 30172 T^{4} - 137236 T^{5} + 1587654 T^{6} - 5420890 T^{7} + 1587654 p T^{8} - 137236 p^{2} T^{9} + 30172 p^{3} T^{10} - 2312 p^{4} T^{11} + 301 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 2 T + 169 T^{2} - 466 T^{3} + 8838 T^{4} - 49142 T^{5} + 4752 T^{6} - 3410288 T^{7} + 4752 p T^{8} - 49142 p^{2} T^{9} + 8838 p^{3} T^{10} - 466 p^{4} T^{11} + 169 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 29 T + 577 T^{2} - 7440 T^{3} + 77552 T^{4} - 620580 T^{5} + 4665518 T^{6} - 33460606 T^{7} + 4665518 p T^{8} - 620580 p^{2} T^{9} + 77552 p^{3} T^{10} - 7440 p^{4} T^{11} + 577 p^{5} T^{12} - 29 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 10 T + 296 T^{2} - 1680 T^{3} + 29384 T^{4} - 51274 T^{5} + 1413461 T^{6} + 2150232 T^{7} + 1413461 p T^{8} - 51274 p^{2} T^{9} + 29384 p^{3} T^{10} - 1680 p^{4} T^{11} + 296 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 370 T^{2} + 478 T^{3} + 62344 T^{4} + 124528 T^{5} + 6478623 T^{6} + 12852964 T^{7} + 6478623 p T^{8} + 124528 p^{2} T^{9} + 62344 p^{3} T^{10} + 478 p^{4} T^{11} + 370 p^{5} T^{12} + p^{7} T^{14} \) | |
| 73 | \( 1 + T + 256 T^{2} + 859 T^{3} + 32742 T^{4} + 174811 T^{5} + 2947121 T^{6} + 17408914 T^{7} + 2947121 p T^{8} + 174811 p^{2} T^{9} + 32742 p^{3} T^{10} + 859 p^{4} T^{11} + 256 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 36 T + 811 T^{2} - 14204 T^{3} + 206123 T^{4} - 2551244 T^{5} + 27484641 T^{6} - 260826920 T^{7} + 27484641 p T^{8} - 2551244 p^{2} T^{9} + 206123 p^{3} T^{10} - 14204 p^{4} T^{11} + 811 p^{5} T^{12} - 36 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 18 T + 476 T^{2} - 6176 T^{3} + 101516 T^{4} - 1046026 T^{5} + 12975605 T^{6} - 108688376 T^{7} + 12975605 p T^{8} - 1046026 p^{2} T^{9} + 101516 p^{3} T^{10} - 6176 p^{4} T^{11} + 476 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + T + 56 T^{2} - 904 T^{3} - 662 T^{4} - 70601 T^{5} + 1164325 T^{6} + 4198720 T^{7} + 1164325 p T^{8} - 70601 p^{2} T^{9} - 662 p^{3} T^{10} - 904 p^{4} T^{11} + 56 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 19 T + 518 T^{2} + 7608 T^{3} + 122496 T^{4} + 1418373 T^{5} + 17665881 T^{6} + 166288352 T^{7} + 17665881 p T^{8} + 1418373 p^{2} T^{9} + 122496 p^{3} T^{10} + 7608 p^{4} T^{11} + 518 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.49501833073950073295555108937, −3.47166484894435208224045761180, −3.39672126865006778250789543537, −3.11696711670881720341494379618, −2.84947731016054190343373487585, −2.77267644785898921927070190251, −2.70054049265471736007692505482, −2.62522475074821575900764056147, −2.44423300392657059698516372069, −2.31760059693971113710993500085, −2.26489789699223027399487612631, −2.12348788764605515880780588915, −2.06809069849301407047582880784, −1.93253548173572737639977307529, −1.76696119525466527432033522043, −1.56643677222108722992660745431, −1.56004097327874124849360826540, −1.27832405874043408672337309414, −1.23875884554833432867328895960, −0.913342428546967808376663630924, −0.74946800334243406449449901453, −0.70243756901828416037622546321, −0.63221794889917267400343486917, −0.41475606253146945851088856952, −0.25073317661328305215889748534, 0.25073317661328305215889748534, 0.41475606253146945851088856952, 0.63221794889917267400343486917, 0.70243756901828416037622546321, 0.74946800334243406449449901453, 0.913342428546967808376663630924, 1.23875884554833432867328895960, 1.27832405874043408672337309414, 1.56004097327874124849360826540, 1.56643677222108722992660745431, 1.76696119525466527432033522043, 1.93253548173572737639977307529, 2.06809069849301407047582880784, 2.12348788764605515880780588915, 2.26489789699223027399487612631, 2.31760059693971113710993500085, 2.44423300392657059698516372069, 2.62522475074821575900764056147, 2.70054049265471736007692505482, 2.77267644785898921927070190251, 2.84947731016054190343373487585, 3.11696711670881720341494379618, 3.39672126865006778250789543537, 3.47166484894435208224045761180, 3.49501833073950073295555108937
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.